The Landscape of L-functions: degree 3 and conductor 1

TL;DR

This paper numerically explores the landscape of degree 3, conductor 1 L-functions, revealing their distribution and density near the origin, and extends theoretical understanding of their existence at small spectral parameters.

Contribution

It provides the first extensive numerical data for degree 3, conductor 1 L-functions and discusses their distribution in a geometric landscape, extending previous theoretical results.

Findings

L-functions are densely clustered near the origin.

No L-functions exist for very small spectral parameters.

Points have smaller density than expected at large spectral parameters.

Abstract

We extend previous lists by numerically computing approximations to many L-functions of degree $d=3$, conductor $N=1$, and small spectral parameters. We sketch how previous arguments extend to say that for very small spectral parameters there are no such L-functions. Using the case $(d,N) = (3,1)$ as a guide, we explain how the set of all L-functions with any fixed invariants $(d,N)$ can be viewed as a landscape of points in a $(d-1)$-dimensional Euclidean space. We use Plancherel measure to identify the expected density of points for large spectral parameters for general $(d,N)$. The points from our data are close to the origin and we find that they have smaller density.

Tables

Table 1. Table 2.1. Γ Γ \Gamma -types γ 𝛾 \gamma and refined signatures δ = ( d + , d − , d 2 ) 𝛿 subscript 𝑑 subscript 𝑑 subscript 𝑑 2 \delta=(d_{+},d_{-},d_{2}) indexing inclusions X γ ⊆ X δ ′ subscript 𝑋 𝛾 subscript superscript 𝑋 ′ 𝛿 X_{\gamma}\subseteq X^{\prime}_{\delta} of connected spaces in degree d ≤ 3 𝑑 3 d\leq 3

Equations

$${\mathcal{L}}_{d,N}={\mathcal{L}}_{d,N}^{\mathrm{alg}}\coprod {\mathcal{L}}_{d,N}^{\mathrm{trans}}.$$

$$P_3=\frac{\sqrt{3}\zeta (3)}{128{\pi }^3}\approx \mathrm{0.0005246.}$$

$$L(s)=\prod _{\mathrm{primes}p}\frac{1}{1-a_p{p}^{-s}+{\overline{a}}_p{p}^{-2s}-p^{-3s}}.$$

$$\mathcal{L}={\mathcal{L}}^{\mathrm{alg}}\coprod {\mathcal{L}}^{\mathrm{trans}}.$$

$$L(s)=\prod _{\mathrm{primes}p}\frac{1}{f_p(p^{-s})}=\sum _{n=1}^{\infty }\frac{a_n}{n^s};$$

$$f_p(x)=1-a_{p}x+\cdots +(-1{)}^d\chi (p)x^d.$$

$$\Lambda (s)$$

$$=\epsilon \overline{\Lambda }(1-s).$$

$${\Gamma }_{\mathbb{R}}(s)={\pi }^{-s/2}\Gamma (s/2),{\Gamma }_{\mathbb{C}}(s)=(2\pi {)}^{-s}\Gamma (s).$$

$${\mu }_j$$

$$\chi (-1)=(-1{)}^{\sum {\delta }_j+\sum (2{\kappa }_k+1)}.$$

$${\Gamma }_{\mathbb{R}}(s+i{\lambda }_1){\Gamma }_{\mathbb{R}}(s+i{\lambda }_2){\Gamma }_{\mathbb{R}}(s+i{\lambda }_3),$$

$${\Gamma }_{\mathbb{R}}(s+i{\lambda }_1){\Gamma }_{\mathbb{R}}(s+1+i{\lambda }_2){\Gamma }_{\mathbb{R}}(s+1+i{\lambda }_3).$$

$${\Gamma }_{\mathbb{R}}(s+\delta +i\lambda ){\Gamma }_{\mathbb{C}}(s+\kappa -i\lambda /2).$$

$$\Lambda (s)g(s)=Q^s\sum _{n=1}^{\infty }\frac{a_n}{n^s}{f}_1(s,n)+\epsilon Q^{1-s}\sum _{n=1}^{\infty }\frac{{\overline{a}}_n}{n^{1-s}}{f}_2(1-s,n).$$

$$Z(5)=$$

$$-0.000023a_7^r-0.000023a_7^i+\cdots -5.0\times 10^{-10}{a}_{11}^r-6.2\times 10^{-10}{a}_{11}^i+\cdots .$$

$$Z(5)=$$

$$+0.000022a_7^r+0.000012a_7^i+\cdots +3.5\times 10^{-10}{a}_{11}^r-1.1\times 10^{-0}{a}_{11}^i+\cdots .$$

$$\begin{array}[]{rcrlrl}\lambda&\approx&&\!\!\!16.9761877662491\!\!\!\!\!\!\\ a_{2}&\approx&-\!\!\!&0.063424433675\!\!\!&+\!\!\!&0.10988282023\,i\\ a_{3}&\approx&&0.63801509281&+\!\!\!&0.50484439411\,i\\ a_{5}&\approx&&0.9093205878&-\!\!\!&0.1141574688\,i\\ a_{7}&\approx&&0.68168545&+\!\!\!&0.02028876\,i\\ a_{11}&\approx&-\!\!\!&0.33243&+\!\!\!&0.12179\,i\\ a_{13}&\approx&-\!\!\!&0.46&-\!\!\!&0.72\,i,\end{array}$$

$$\sum_{\gamma}f(\gamma)=\hat{f}(0)\cdot\text{(log $N$)}+\int f(t)\frac{\Gamma^{\prime}}{\Gamma}\text{-terms}\,dt+\sum_{n}\hat{f}\bigl{(}\frac{\log n}{2\pi}\bigr{)}\text{coeffs}.$$

$$\displaystyle\Gamma_{\mathbb{R}}(s+i\lambda)\ \mbox{ contributes }\$$

$$\displaystyle\Gamma_{\mathbb{R}}(s+1+i\lambda)\ \mbox{ contributes }\$$

$$\displaystyle\Gamma_{\mathbb{C}}(s+\kappa+i\lambda)\ \mbox{ contributes }\$$

$$f_{ϵ}(x)=\prod _{z\in Z_{ϵ}}(x-z)\in \mathbb{R}[x].$$

$$f(x)=f_{+}(x)f_{-}(x)f_2(x)=x^d+c_2{x}^{d-2}+\cdots +c_{d-1}x+c_d.$$

$$|\mathcal{L}(U,N)|\approx {\mu }_d(U){\nu }_d(N),$$

$$\sum _{N\le x}|\mathcal{L}(U,N)|\sim \sum _{N\le x}{\mu }_d(U){\nu }_d(N).$$

$$|t{|}_{+}$$

$${\mu }_d$$

$$X_{+++}$$

$$X_{+--}$$

$$X_{-c}$$

$${\mu }_d^{\prime }=\frac{P_d}{2^{d_1}}\prod _{i<j}|z_i-z_j|d{\lambda }_1\cdots d{\lambda }_{d_1+d_2-1}d{\kappa }_1\cdots d{\kappa }_{d_2}.$$

Full text

The authors thank Ralf Schmidt and Akshay Venkatesh for helpful conversations. A portion of this work was supported by the National Science Foundation.

1. Introduction

1.1. Overview via pictures

Consider the set ${\mathcal{L}}_{d,N}$ of all automorphic L-functions of degree $d$ and conductor $N$, as defined in Section 2. It decomposes into well-defined subsets of algebraic and transcendental L-functions,

[TABLE]

The algebraic part is of particular interest because it conjecturally agrees with the set of analytically-normalized motivic L-functions of the given degree and conductor. But the transcendental part is of great interest too, in part because for $d\geq 3$ it is much larger than the algebraic part. Tabulation efforts, such as the LMFDB, have focused mainly on ${\mathcal{L}}^{\rm alg}_{d,N}$ and, via classical Maass wave forms, ${\mathcal{L}}_{2,N}$. In this paper, a sequel to [6], we aim to bring more computational attention to the very large sets ${\mathcal{L}}^{\rm trans}_{d,N}$ for $d\geq 3$. We focus on the first case $(d,N)=(3,1)$.

We think about the sets ${\mathcal{L}}_{d,N}$ visually, with L-functions being represented by points in regions of coordinatized $(d-1)$-dimensional Euclidean spaces. Each of these L-points has a natural attached multiplicity, usually $1$. A region together with its collection of L-points is a landscape. We have two organizational schemes. One breaks ${\mathcal{L}}_{d,N}$ into $\lfloor(d+2)^{2}/4\rfloor$ disjoint parts and places each part in its own parameter landscape. The other places all the points in a single coefficient landscape, where the region is all of $\mathbb{R}^{d-1}$. The names come from the nature of the coordinates, which are real and imaginary parts of spectral parameters in the first organizational scheme and coefficients $c_{2}$, …, $c_{d}$ of a polynomial in the second.

For our case $(d,N)=(3,1)$, Figure 3.1 and Figure 3.2 each draw a parameter landscape, while Figure 3.3 superimposes two similar parameter landscapes. The two remaining parameter landscapes for $(d,N)=(3,1)$ are theoretically obstructed from having L-points, a special feature of the small conductor $N=1$, and hence not drawn. Figure 5.2 draws the coefficient landscape, with the contribution of each parameter landscape being indicated via color. It becomes computationally more difficult to find L-functions as one goes further away from the origin in a given parameter space, and all our figures are of course partial in that they only show the known L-points. Also our methods are numerical, so some L-points could conceivably be incorrect, although we think this is unlikely. We encourage the reader to turn to these four figures, because they give an accurate first sense of the content of this paper.

We are interested in understanding the landscapes both computationally, as just indicated, and theoretically. The foundational theoretical concepts are Plancherel measure $\mu_{d}$ and its canonical approximation $\mu_{d}^{\prime}$. Attention to these concepts is a significant advance in this paper beyond the results of [6]. The very similar measures $\mu_{d}$ and $\mu_{d}^{\prime}$ are indicated by contour plots on our three figures of parameter landscapes. They are not indicated directly on the coefficient landscape because they do not need to be: $\mu^{\prime}_{d}=P_{d}\,dc_{2}\cdots dc_{d}$ is exactly Euclidean. The meaning of $\mu_{d}$ is that it conjecturally governs the distribution of L-points in a precise asymptotic sense. We combine three formulas from the literature to identify the Plancherel constants $P_{d}$. The relevant one for our case is

[TABLE]

While our new data is only in the $(d,N)=(3,1)$ case, our text works with general $(d,N)$ as it is aimed at supporting future numerical investigations.

1.2. Content of the sections

Section 2 defines the sets ${\mathcal{L}}_{d,N}$, reviews standard material about L-functions, and defines the $\lfloor(d+2)^{2}/4\rfloor$ receiver regions for the parameter landscapes. Section 3 populates the landscapes with L-points, following the methods of [6]. Section 4 offers reminders that each individual L-function is interesting and sketches how one can prove that parameter landscapes where the conductor $N$ is small enough have an L-point-free region near their origin.

Section 5 discusses the purely algebraic passage from parameter landscapes to the coefficient landscape by means of symmetric functions. Section 6 gives the Plancherel theory including the identification of the constants $P_{d}$. Our data gives points at roughly half the asymptotic density $P_{d}$. We explain by comparison with the well-understood $d=2$ case that this is an expected shortfall; there should be as yet undiscovered secondary terms which will make theory better match the data. Finally Section 7 gives two independent complements. One compares our data with the $p$-adic Plancherel measures on the class space of $\operatorname{SU}_{3}$, finding again the need for secondary terms. The other discusses the extent to which transcendental L-functions outnumber algebraic L-functions in degrees $\geq 3$.

1.3. Availability of data

The L-functions in ${\mathcal{L}}_{3,1}$ have the form

[TABLE]

For each drawn L-point, its coordinates in the landscape and the complex numbers $a_{p}$ to their computed precision are available on GitHub at [8]. Earlier data, dating back to [6, 3, 4], is already conveniently available on the LMFDB for $(d,N)\in\{(3,1),(3,4),(3,9),(4,1)\}$. We plan to systematically extend this section of the LMFDB, because much more is within computational reach.

2. Background on L-functions

Here we review L-functions with an initial focus on the fundamental decomposition of all automorphic L-functions into the two types,

[TABLE]

Whether an automorphic L-function is algebraic or transcendental depends only on the $\Gamma$-factor in its functional equation and our second main focus is the space $X_{d}$ of possible $\Gamma$-factors in degree $d$, and various related spaces.

2.1. The set of automorphic L-functions ${\mathcal{L}}$

As a terminological catch-all, define an L-function to be a Dirichlet series with Euler product converging in some right half plane:

[TABLE]

here the $f_{p}(x)=1+\cdots$ are required to be polynomials in $\mathbb{C}[x]$ having a maximal degree $d$. For example, the Riemann zeta function $\zeta(s)$ has $f_{p}(x)=1-x$ for all $p$ and degree $1$. The class of L-functions just defined is not at all interesting in its entirety. However it allows us to conveniently talk about automorphic, motivic, and our experimental L-functions.

An automorphic L-function for this paper is an L-function associated in the standard way to a balanced tempered-at-$\infty$ cuspidal automorphic representation of the adelic group $GL_{d}(\mathbb{A})$. We do not need to enter into the very complicated automorphic theory, as we will just be using known properties of these L-functions. A list of known properties which moreover conjecturally characterizes automorphic L-functions is given in [9]. Here we review what we need.

Associated to an automorphic L-function is a conductor $N\in\mathbb{Z}_{\geq 1}$. Also associated is a central character $\chi$, which is a Dirichlet character of conductor $N$. So $\chi:\mathbb{Z}/N\rightarrow\mathbb{C}$ is a multiplicative homomorphism, making $\chi(p)=0$ for the primes $p$ dividing $N$. The L-functions coming from $GL_{d}(\mathbb{A})$ have degree $d$, as

[TABLE]

The Riemann zeta function is automorphic, and has a meromorphic continuation to the whole plane with a unique pole at $s=1$. All other automorphic L-functions are entire. The completed L-function and its functional equation take the form

[TABLE]

Here the bar indicates Schwarz reflection, so that $\overline{L}(s)=\overline{L(\overline{s})}=\sum_{n}\overline{a}_{n}n^{-s}$ is the dual of $L$. Also $\Gamma_{\mathbb{R}}$ and $\Gamma_{\mathbb{C}}$ are the normalized $\Gamma$-functions,

[TABLE]

The double product of $\Gamma$-functions in (2.3) is the $\Gamma$-factor of $L$, to be discussed in more detail in the next subsection. The sign $\varepsilon$ is a complex number on the unit circle. The $d_{i}\in\mathbb{Z}_{\geq 0}$ satisfy $d_{1}+2d_{2}=d$ and form the signature $(d_{1},d_{2})$ of $L$.

2.2. $\Gamma$-factors

Much of this paper is driven by the nature of $\Gamma$-factors, and so the terms introduced in this section are particularly important. The spectral parameters $\mu_{j}$ and $\nu_{k}$ in (2.3) are written in terms of their real and imaginary parts as

[TABLE]

They are constrained by the balanced condition in our definition of automorphic L-function, $\sum\lambda_{j}+2\sum\beta_{k}=0$. They are constrained also by the tempered-at-$\infty$ condition, which requires $\delta_{j}\in\{0,1\}$ and $\kappa_{k}\in\{\frac{1}{2},1,\frac{3}{2},2,\dots\}$. There is no ambiguity in $d_{1}$ and $d_{2}$ coming from the duplication formula $\Gamma_{\mathbb{C}}(s)=\Gamma_{\mathbb{R}}(s)\Gamma_{\mathbb{R}}(s+1)$ because $\kappa_{k}=0$ is disallowed. Important for us is the refined signature $(d_{+},d_{-},d_{2})$, where $d_{+}$ and $d_{-}$ are the number of $j$ with $\delta_{j}=0$ and $\delta_{j}=1$ respectively. Henceforth, we write $\lambda_{d_{1}+k}$ rather than $\beta_{k}$. To remove ambiguities, we require that the $\mu_{j}$ and $\nu_{k}$ are both increasing with respect to the standard lexicographical order on $\mathbb{C}$.

The $\Gamma$-type of a $\Gamma$-factor is the list of $\delta_{j}$ followed by the list of $2\kappa_{k}$. For example, a degree seven L-function might have $\Gamma$-type $(0,0,1,1,1;5)$. Following LMFDB conventions, we normally write $\Gamma$-types by prefixing each $\delta_{i}$ with an $r$ and each $2\kappa_{k}$ with a $c$, as in $r0r0r1r1r1c5$. A $\Gamma$-type is called even if all $2\kappa_{k}$ are even. It is called odd if $d_{1}=0$ and all $2\kappa_{k}$ are odd. Thus our example $\Gamma$-type is neither odd nor even.

There is an important constraint on the spectral parameters in terms of the central character:

[TABLE]

Always $\chi(-1)=\epsilon_{\infty}^{2}$, where $\epsilon_{\infty}\in\{1,i,-1,-i\}$ depends multiplicatively on the individual Gamma factors in $\eqref{Ldef}$, in the way presented in Table 2.1. There are two simplifications when $N=1$. First, $\chi(-1)=1$, allowing one to disregard half the possible spectral parameters, namely those that fail (2.6). Second $\varepsilon=\epsilon_{\infty}$; this replaces an unknown quantity with a known one in the searches described in §3.5.

2.3. The decomposition ${\mathcal{L}}={\mathcal{L}}^{\rm alg}\coprod{\mathcal{L}}^{\rm trans}$

By definition, an automorphic L-function is algebraic if its spectral parameters are real and its $\Gamma$-type has a parity. The remaining automorphic L-functions are transcendental. The next two paragraphs explain how this innocuous-looking decomposition is expected to be a very sharp dichotomy.

A motivic L-function $L$ comes from an elaborate process that starts by counting points on a fixed algebraic variety over varying finite fields. It has an associated $\Gamma$-factor coming from Hodge theory. We assume a primitivity condition corresponding to focusing attention on an irreducible motive. After switching to the analytic normalization, a change of variables of the form $s\rightarrow s+\frac{w}{2}$ for $w$ an integer, a fundamental and widely-believed conjecture says that motivic L-functions are automorphic. By the nature of their $\Gamma$-factors, they would have to be in ${\mathcal{L}}^{\rm alg}$. A conjectural converse says that all L-functions in ${\mathcal{L}}^{\rm alg}$ come from motives in this way.

For a motivic L-function, even in its analytic normalization, all the coefficients $a_{n}$ are algebraic numbers. In contrast, for a transcendental L-function one expects that all the nonzero $\lambda_{j}$ and almost all the nonzero $a_{p}$ are transcendental numbers. Assuming that all algebraic L-functions are motivic, this conjecture translates to a contrast between ${\mathcal{L}}^{\rm alg}$ and ${\mathcal{L}}^{\rm trans}$. The contrast also explains the names of the summands.

We refer the reader again to [9] for more details on everything discussed so far. One subtlety worth highlighting concerns Axioms 4a and 4b on the list of axioms in Section 2.1 of [9]. We are imposing Axiom 4a by our “temperedness-at-$\infty$” condition, because it is necessary for our points-in-landscapes presentation. The Selberg conjecture says that we would not enlarge our set ${\mathcal{L}}$ if we removed this condition. We are not imposing Axiom 4b, which corresponds to a “temperedness at all finite primes $p$” condition, because it is not necessary for our presentation. Here the Ramanujan conjecture says that we would not decrease our set ${\mathcal{L}}$ if we added this condition.

2.4. The constructional problem

Consider now the problem of explicitly writing down algebraic L-functions, say in an approximate form $\sum_{n=1}^{c}a_{n}n^{-s}$. One can do this easily by purely automorphic methods only in very limited circumstances. For example, consider automorphic L-functions with $\Gamma$-factor $\Gamma_{\mathbb{C}}(s+1/2)^{g}$ and $\chi$ the trivial character. Then for $g=1$, automorphic methods are successful: one can find a corresponding holomorphic newform of weight $2$ on some $\Gamma_{0}(N)$. For $g\geq 2$, it is not immediately clear by automorphic methods that $\Gamma_{\mathbb{C}}(s+1/2)^{g}$ even actually arises, and explicitly computing a particular $\sum_{n=1}^{c}a_{n}n^{-s}$ even for small cutoff $c$ seems problematic. In contrast, writing down a genus $g$ curve over $\mathbb{Q}$ and computing many $a_{n}$ for it is much more straightforward.

Consider now the same constructional problem but now for transcendental L-functions. Purely automorphic methods seem even less promising in this context. In this paper, we do what seems like the best that can be currently done: we produce experimental L-functions $\prod_{p}f_{p}(x)^{-s}$ having only finitely many non-trivial factors. Their purpose is to approximate actual automorphic L-functions. At present, like in typical instances of using motivic L-functions as in the last paragraph, the evidence for the existence of a corresponding automorphic L-function is strong, but not conclusive.

Having carefully made distinctions, we will now generally be more brief with our language: “L-function” in theoretical contexts will mean automorphic L-function; “L-function” in the context of our searches will mean experimental L-function. One would like to populate our landscapes with actual automorphic L-points, but we are of course populating them with experimental L-points.

2.5. Spaces of generalized $\Gamma$-factors

The material of this subsection is completely elementary. However it is not standard, but rather is particular to our concerns in this paper. To define the Euclidean regions receiving L-points, it is necessary to allow generalized $\Gamma$-factors by relaxing the half-integrality condition on the $\kappa_{k}$ to $\kappa_{k}\in(0,\infty)$. Let $X^{\prime}_{d_{-},d_{+},d_{2}}$ be the set of $\Gamma$-factors with refined signature $(d_{-},d_{+},d_{2})$. Because of our ordering convention, the $\lambda_{j}$ and $\kappa_{k}$ are well-defined real-valued functions on $X^{\prime}_{d_{-},d_{+},d_{2}}$. Because of the balanced condition, the functions $(\lambda_{1},\dots,\lambda_{d_{1}+d_{2}-1},\kappa_{1},\dots,\kappa_{d_{2}})$ render $X^{\prime}_{d_{-},d_{+},d_{2}}$ a full-dimensional cone in $\mathbb{R}^{d-1}$ and thus a connected topological space. Let $X^{\prime}_{d_{1},d_{2}}$ be the disjoint union of the $X^{\prime}_{d+,d-,d_{2}}$ with $d_{1}=d_{+}+d_{-}$. Let $X^{\prime}_{d}$ likewise be the disjoint union of the $X^{\prime}_{d_{1},d_{2}}$ with $d=d_{1}+2d_{2}$. So clearly $X^{\prime}_{d_{1},d_{2}}$ has $d_{1}+1$ connected components. As reported in the introduction, the number of connected components of $X_{d}^{\prime}$ works out to the “quarter square” $\lfloor(d+2)^{2}/4\rfloor$. As an alternative notation for $X^{\prime}_{d_{+},d_{-},d_{2}}$ we use repetition, as in $X^{\prime}_{2,3,1}=X^{\prime}_{++---c}$. The possible spaces $X^{\prime}_{d_{+},d_{-},d_{2}}$ with $d\leq 3$ are listed in this alternative notation in Table 2.1.

The subspaces of all the above spaces consisting of actual $\Gamma$-factors are denoted by removing the prime. The connected components of $X_{d}$ are indexed by their $\Gamma$-types, as discussed above. All the components of $X_{d_{1},d_{2}}$ have dimension $d_{1}+d_{2}-1$, there being infinitely many components exactly when $d_{2}>0$. One of the virtues of LMFDB notation for $\Gamma$-types is that it lets one easily distinguish the connected spaces $X^{\prime}_{d_{+},d_{-},d_{0}}$ from the connected components of their subspaces $X_{d_{+},d_{-},d_{2}}$. As an example, $r0r0r1r1r1c5\subset X^{\prime}_{++---c}$ is an inclusion of a connected five-dimensional space into a connected six-dimensional space. In general, the notational transition is made by $r0\rightarrow+$, $r1\rightarrow-$, and replacing all the $2\kappa_{k}$’s by $c$’s. All possibilities for $d\leq 3$ are listed in Table 2.1.

2.6. Sources

While we do not need the full automorphic theory, we will reference automorphic forms in classical contexts. Very simply, it is known that ${\mathcal{L}}_{1,N}={\mathcal{L}}^{\rm alg}_{1,N}$ is the set of Dirichlet L-functions of conductor $N$. In the next dimension, ${\mathcal{L}}_{2,N}$ is conjecturally the set $L(f,s)$ of L-functions coming from newforms on $\Gamma_{1}(N)$. Holomorphic forms and Maass forms with Laplacian eigenvalue $\frac{1}{4}$ give L-functions in ${\mathcal{L}}_{2,N}^{\rm alg}$. Maass forms with Laplacian eigenvalue $\lambda>\frac{1}{4}$ give L-functions in ${\mathcal{L}}_{2,N}^{\rm trans}$. There are not expected to be Maass forms with Laplacian eigenvalue $<\frac{1}{4}$ because their L-functions would be counterexamples to the Selberg conjecture; if such L-functions exist, they would not be in our ${\mathcal{L}}_{2,N}$. For $d\geq 3$, one is already out of the classical setting. Table 2.1 summarizes the situation.

3. The parameter landscapes for L-functions with $(d,N)=(3,1)$

Here we draw the three pictures promised in the introduction and discuss how we found the L-functions corresponding to the L-points in the pictures.

3.1. The landscape of L-functions with $\Gamma$-type $r0r0r0$ in

the cone $X_{+++}$

This is the landscape studied in [6] and we roughly double the number of L-points previously available in it. As explained in §2.5, the L-functions considered here have $\Gamma$-factor of the form

[TABLE]

with $\lambda_{j}\in\mathbb{R}$ and $\lambda_{1}+\lambda_{2}+\lambda_{3}=0$. The general prescription of §2.2, useful for uniform theoretical discussions, is in this case to require $\lambda_{1}\leq\lambda_{2}\leq\lambda_{3}$, and use $\lambda_{1}$ and $\lambda_{2}$ as coordinates. Instead, we will be consistent with [6] and order by $\lambda_{3}\leq\lambda_{2}\leq\lambda_{1}$, still using $\lambda_{1}$ and $\lambda_{2}$. This difference in conventions is unimportant: it is a question of choosing a fundamental domain for a Euclidean plane with an action by the symmetric group $S_{3}$.

There is a second redundancy not emphasized in the previous section: an L-function and its dual determine each other by complex conjugation. To remove this redundancy, again following [6], we require $\lambda_{2}\geq 0$. Accordingly, Figure 3.1 draws only half of the landscape; the other half is obtained by reflecting in the $\lambda_{1}$-axis. So the shaded region is a fundamental domain for the action of a dihedral group $D_{6}=S_{3}\times S_{2}$. It is naturally one-twelfth of its plane, but our symmetry-breaking choice to emphasize $\lambda_{1}$ and $\lambda_{2}$ obscures this fact.

There are six L-points in Figure 3.1 on the $\lambda_{1}$-axis. They arise as the symmetric squares of L-functions of the classical Maass forms in appearing as the six rightmost colored L-points in Figure 6.2. The coordinates $(\lambda_{1},0)$ in Figure 3.1 and $(\lambda)$ in Figure 6.2 are related by $\lambda_{1}=2\lambda$. It is known that the non-algebraic self-dual L-functions in ${\mathcal{L}}_{1,3}$ arise in this way.

A careful look at Figure 3.1 reveals striations running along lines of approximate slope [math] and $1$. It would be interesting to find a theoretical explanation for this pattern.

3.2. The landscape of L-functions with $\Gamma$-type $r0r1r1$ in

the cone $X_{+--}$

Our second landscape is populated with L-functions with $\Gamma$-factor of the form

[TABLE]

Here $\lambda_{2}$ and $\lambda_{3}=-\lambda_{1}-\lambda_{2}$ are playing the same role and so we normalize by requiring $\lambda_{2}\geq\lambda_{3}$. The bottom boundary of the drawn region comes from the equation $\lambda_{2}=\lambda_{3}$, i.e. $2\lambda_{1}+\lambda_{2}=0$. As $\lambda_{1}$ is playing a different role from the other two $\lambda_{i}$, we can remove the redundancy of duality by restricting attention to $\lambda_{1}\geq 0$. So the highlighted region is a fundamental domain for a group of the form $C_{2}\times C_{2}$, and so really should be thought of as one-quarter of the full plane.

In contrast with the previous subsection, where all L-functions have sign $1$, here all L-functions have sign $-1$, by Table 2.1. These signs are important in our search for L-functions. However, the sign being $-1$ does not at all force central vanishing of the L-functions, since they are not self-dual. Applying general principles to our cases here, we would expect that $L(1/2)\neq 0$ for all any $L$ in any ${\mathcal{L}}^{\rm trans}_{3,N}$.

3.3. The landscape of $r0c2\kappa$ lines in $X^{\prime}_{+c}$ and

the landscape of $r1c2\kappa$ lines in $X^{\prime}_{-c}$

Our third case illustrates typical behavior for arbitrary $d$ better than the first two cases. Here the L-functions have $\Gamma$-factors of the form

[TABLE]

We work in the $(\kappa,\lambda)$ plane. For $\tau\in\{+,-\}$, the set $X^{\prime}_{\tau c}$ is identified with the half-space where $\kappa>0$. Following the general prescription, a point in $X^{\prime}_{+c}$ is a generalized $\Gamma$-factor (3.3) with $\delta=0$; similarly, points in $X^{\prime}_{-c}$ index generalized $\Gamma$-factors (3.3) with $\delta=1$. To remove the redundancy given by duality, we draw only the quadrant where also $\lambda\geq 0$.

The spaces $X_{\tau c}$ of actual $\Gamma$-factors come from imposing the condition that $\kappa\in\frac{1}{2}\mathbb{Z}_{\geq 1}$. Because of the central character condition (2.6) and the identification of $\chi(-1)$ on Table 2.1, L-points in $X_{-c}$ can only have integral $\kappa$ coordinates while L-points in $X_{+c}$ can only have half-integral $\kappa$. It is appropriate to superimpose the two landscapes in one figure, because the formula (6.3) for Plancherel measure $\mu$ on $X_{\tau 0}$ and the formula (6.4) for approximate Plancherel measure $\mu^{\prime}$ on $X^{\prime}_{\tau 0}$ are independent of $\tau\in\{-,+\}$. This agreement of measures holds always between the spaces indexed by $(a,b,d_{2})$ and $(b,a,d_{2})$. Thus $X_{+++}$ from §3.1 would be naturally superimposed with $X_{---}$ and $X_{+--}$ from §3.2 would be naturally superimposed with $X_{-++}$. To repeat a point from the introduction, it is only because $N=1$ that each of the latter spaces do not have L-points.

3.4. The algebraic L-points

There is only one algebraic L-point in our three figures, it being $(11,0)$ in Figure 3.3. However for the analogous figures with increasing $N$, algebraic L-points start to appear as follows.

Any space $X_{d_{+},d_{-},0}$ has exactly one algebraic point, the origin $(0,\dots,0)$. All the L-functions mapping to $(0,\dots,0)$ are expected to come from the continuous representations of $\mbox{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ into $\operatorname{GL}_{d}(\mathbb{C})$ for which complex conjugation has eigenvalues $1$ and $-1$ with the respective multiplicities $d_{+}$ and $d_{-}$. The LMFDB catalogs many examples, with $d_{+}-d_{-}$ given in the column $\chi(c)$. Self-dual L-functions can be searched by making the Frobenius-Schur indicator $1$. For the four spaces $X_{+++}$, $X_{++-}$, $X_{+--}$, and $X_{---}$, the smallest conductors appearing are respectively $1957$, $283$, $229$, and $2828$. The images of the representations are the groups $S_{4}$, $S_{4}$, $S_{4}$, and $S_{4}\times S_{2}$, and the L-functions all come from Dirichlet twists of symmetric squares of rank two L-functions. Non-self-dual L-functions have Frobenius-Schur indicator [math]. For the first three spaces, the smallest conductors appearing are respectively $8473$, $1228$, and $8100$, from groups $C_{3}\wr S_{3}$, $(C_{3}^{2}\!:\!C_{3})\!:\!C_{3}$, and $(C_{3}^{2}\!:\!C_{3})\!:\!C_{3}$. As of this writing, the LMFDB does not have an example with eigenvalues $---$, but many can be obtained via twisting a $+++$ example with an odd Dirichlet character.

For the spaces $X_{\tau c}$, the algebraic points are $(\kappa,0)$ for $\kappa$ a positive integer. In direct analogy with a paragraph in §3.1, modular forms of weight $k=\kappa+1$ give a large supply, via symmetric squares and subsequent Dirichlet twists. The aforementioned algebraic point $(11,0)$ in Figure 3.3 comes from the Ramanujan function $\Delta\in S_{12}(1)$. The very low L-point on the $\kappa=13$ line has $\lambda\approx 0.1660$. The fact that the corresponding L-function is not self-dual is seen clearly also in its first coefficients: $a_{2}\approx 0.0256-0.7997i$ and $a_{3}\approx-0.5418+1.2019i$ are far from real. §7.3 discusses the part of ${\mathcal{L}}^{\rm alg}_{3,N}$ with $\kappa>0$ that does not come from modular forms.

3.5. The search method

Our method for locating L-points in landscapes was explained in [6]. We are working with more general $\Gamma$-factors, but the basic method has not changed. We give a quick summary here.

The starting point is the “approximate functional equation”

[TABLE]

Here $Q=\sqrt{N}$ and $f_{1}$ and $f_{2}$ are integrals involving a test function $g(s)$ and the $\Gamma$-factors. The main idea is that the test function $g(s)$ is a free parameter. Thus, by evaluating the L-function in different ways but at the same point $s$, one obtains an equation in the Dirichlet coefficients $a_{n}$. Besides the linear equations coming from (3.4) there are also non-linear equations coming from the Euler product (1.3), reducing the set of unknowns to the prime-indexed coefficients $a_{p}$. If the functional equation parameters correspond to an actual L-function, then that system of equations will be consistent. If the parameters do not correspond to an actual L-function, then the ability to quantify the inconsistency of the system can be turned into a method of searching for the correct parameters. The solution to the consistent system will be the correct $a_{p}$’s.

To illustrate the idea, we sketch how we found the L-point with coordinates $(\kappa,\lambda)=(2.5,16.97618\dots)$ on Figure 3.3. We work with an associated real-valued Z-function with $Z(t)$ directly related to $L(\frac{1}{2}+it)$ as discussed in §4.1 below. Consider a hypothetical (but actually non-existent) L-point with exact spectral parameters say $(\kappa,\lambda)=(\frac{5}{2},16.97)$. Equation (3.4) with $g(s)=1$ says

[TABLE]

With $g(s)=e^{is/2}$, the equation (3.4) says

[TABLE]

The notation $a_{n}^{r}$ and $a_{n}^{i}$ refers to the real and imaginary parts of the Dirichlet coefficient $a_{n}$, respectively. The coefficients come from the integrals $f_{1}$ and $f_{2}$ and can easily be obtained to hundreds of digits of precision.

The equality of the two expressions for $Z(5)$ gives one equation in the Dirichlet coefficients. Choosing other test functions likewise gives other equations. The rapid decrease in the contributions of the coefficients enables one to truncate the expression to obtain an equation in finitely many variables. We can create more equations than unknowns and then solve a sub-system with equal number of equations and unknowns. Typically that system has a solution, and often it has several (recall that it is a nonlinear system because of the relations coming from the Euler product). Plugging these solutions into the unused equations (which we call “detectors”) gives a measure for the consistency of the overdetermined system.

We repeat the process for another hypothetical (but also non-existent) L-point, say with $(\kappa,\lambda)=(\frac{5}{2},16.98)$. By comparing the residuals from the detectors at the two hypothetical L-points, we can use the secant method to determine a more accurate estimate for the coordinates of an actual L-point. When it is successful, our values for $\lambda$ and the $a_{p}$ are improved, typically obtaining one or two additional decimal places of accuracy on each iteration. We generally stop when we are confident that we have computed the spectral parameters to at least a dozen digits of accuracy. In this example, we stopped at

[TABLE]

The same computations enable us to evaluate the L-function accurately, in this case finding $Z(5)\approx-0.03656426734$. This $Z$-function is the bottom one graphed in Figure 4.1.

An important point is that the test functions need to be chosen sufficiently different from each other to make the system well-conditioned. This is difficult to do when the spectral parameters are large and is a limitation of the method. Another interesting point is that we are forced to work to high precision, typically to more than 50 decimal places of accuracy, because the system is ill-conditioned and many digits of precision are lost in the calculations. Again for more details, see [6].

3.6. Rigor and completeness

While our methods never actually establish the existence of L-functions, it is worth emphasizing that our methods are capable of producing rigorous statements. For example, they can prove that there are no L-functions with spectral parameters within certain regions. Similarly, suppose one knew somehow from some external reason that a given region contains exactly one L-point. Then our methods can rigorously identify small intervals guaranteed to contains the correct coordinates of the L-point and the correct real and imaginary parts of some initial coefficients $a_{p}$.

In practice, the step in our search process where we are most likely to miss actual L-functions is in the initial scan of a region to find likely candidates. It can happen for example that search parameters which seem to work well in one region miss L-functions in a nearby region. As one leaves the origin and crosses contour lines in any of Figures 3.1, 3.2, and 3.3, we expect the percentage of L-functions that our program has found generally decreases. We expect that we have found almost all L-points up through the first few contours, but it is surely a much smaller percentage towards the end.

4. Zero-free and L-point-free regions

Each transcendental L-function should be viewed as just as remarkable an analytic object as an algebraic L-function, even though it is much harder to access computationally. We begin by considering zero-free intervals on the critical line of a given L-function and conclude by considering L-point-free regions in landscapes of L-functions. We sketch how these two topics are parallel in nature, as the explicit formula can be used to establish such regions of each type.

4.1. Three L-functions and their low-lying zeros

Every L-function has an associated Z-function, characterized up to sign by $|L(\frac{1}{2}+it)|=|Z(t)|$, with $Z(t)$ real and smooth for $t\in\mathbb{R}$. To fix the sign, we follow the LMFDB’s convention of requiring that $Z(t)$ be positive for $t$ a sufficiently small positive number. Figure 4.1 graphs the Z-function behind an L-point on each of the three figures from the previous section.

The Riemann Hypothesis for an L-function is equivalent to the assertion that the nontrivial zeros of the corresponding Z-function are real. The plots in Figure 4.1 make it appear that the Riemann Hypothesis is not true for these three L-functions. Specifically, a positive local minimum or a negative local maximum is caused by non-real zeros of the Z-function, or, equivalently, zeros off the critical line for the L-function. But in fact these local extrema do not point to counterexamples, because the Riemann Hypothesis is a statement about the nontrivial zeros of the L-function. The trivial zeros occur at the poles of the $\Gamma$-factors, and it is these trivial zeros which are causing the behavior which seems surprising at first glance. This phenomenon is difficult to see for algebraic L-functions where all the trivial zeros occur on the negative real axis.

Figure 4.2 shows how the trivial zeros appear to “push aside” the nearby zeros on the critical line. The L-functions have been chosen to all have trivial zeros at height approximately $17$ to facilitate comparison. The trivial zeros at this height come from a gamma factor $\Gamma_{\mathbb{R}}(s+\delta+i\lambda)$ with $\lambda\approx-17$. The pushing is stronger in the first and third cases because the trivial zeros are one step closer: $\delta=0$ in these cases, while $\delta=1$ in the second case.

4.2. The explicit formula explains zero-free intervals

The pushing-aside phenomenon was first observed in [23]. As shown by [2] it can be rigorously established using Weil’s explicit formula. This formula in cartoon form says

[TABLE]

Here $f$ is a test function, $\hat{f}$ is its Fourier transform, $N$ is the conductor of the given L-function, and the sum on the left is over the zeros. The argument in [2] chooses $f$ to be non-negative, so the left side of (4.1) can be interpreted as the weighted count of the zeros within the support of $f$. The test function is chosen to have support in an interval $I$ containing the heights of the trivial zeros in question. Here the $\Gamma^{\prime}/\Gamma$ terms on the right are negative, the effect being stronger when the zeros are closer to the critical line. In this way, with the conductor small enough as well, one can prove certain intervals $I$ indeed contain no zeros.

4.3. The explicit formula and the approximate functional equation explain L-point-free regions

A striking feature of Figures 3.1, 3.2, and 3.3 is that there are no L-points at all in certain large regions. Such an L-point-free region was established for the case of Figure 3.1 in [19] using the explicit formula (4.1). The L-point-free region was enlarged in [6] by using the approximate functional equation (3.4). See Figure 1 in [6].

The basic idea behind the proof in [19] is that for a point $(\lambda_{1},\lambda_{2})$ in the region in question one can choose a test function $f$ taking only nonnegative values in (4.1) so that the right side of (4.1) is negative. But the left side of (4.1) is nonnegative and the conclusion is that no L-function with coordinates at $(\lambda_{1},\lambda_{2})$ can exist. This same general method has been applied in many contexts. For example, simply taking $(\lambda_{1},\lambda_{2})=(0,0)$ one is saying that certain L-functions coming from number fields can only exist if conductors are large enough [13]. The method, and the more powerful exclusionary techniques of [6], can be extended to establish L-point-free regions of other landscapes.

For the discussion in §6.5 and §6.6 we are interested in how the L-point-free region varies with the parameter space. For example, consider the two different parameter spaces which are superimposed in Figure 3.3, for which the Plancherel measures are exactly the same. Observationally, the lowest purple points on lines where $\kappa$ is integral are lower than the lowest green points on lines where $\kappa$ is half-integral. Accordingly, we expect a larger zero-free region in $X_{+c}$ than in $X_{-c}$. From the viewpoint of the explicit formula, the situation is similar to the contrast between the three cases in Figure 4.2. Namely the factor $\Gamma_{\mathbb{R}}(s+i\lambda)$ gives a larger negative contribution to the $\Gamma^{\prime}/\Gamma$-term in (4.1) than the factor $\Gamma_{\mathbb{R}}(s+1+i\lambda)$ does.

5. Coefficient space

Here we treat general degrees $d$ and explain how the union $X^{\prime}_{d}$ of all the connected parameter spaces $X^{\prime}_{d_{+},d_{-},d_{2}}$ maps finite-to-$1$ to a single coefficient space $Y^{\prime}_{d}=\mathbb{R}^{d-1}$. Information is lost, because of the failure of injectivity, but many parameter landscapes, like Figures 3.1, 3.2, and 3.3 are combined into a single coefficient landscape, like Figure 5.2.

5.1. Spectral parameters as three multisets of complex numbers

An equivalent way to give spectral parameters is to give multisets $Z_{+}$, $Z_{-}$, and $Z_{2}$ of complex numbers in a slightly different way, as follows:

[TABLE]

We call the triple $(Z_{+},Z_{-},Z_{2})$ a dot diagram because we think of it visually as illustrated by Figure 5.1.

So the space $X_{d}$ of allowed spectral parameters in degree $d$ is identified with the space of dot diagrams having $d$ points, counting multiplicities. Repeating material from §2.5 in the new language, $X_{d}$ decomposes as the disjoint union of the subspaces $X_{d_{1},d_{2}}$ parameterizing dot diagrams with $d_{2}$ complex conjugate pairs of non-real roots. Each $X_{d_{1},d_{2}}$ then decomposes into the disjoint union of the subspaces $X_{d_{+},d_{-},d_{2}}$ parametrizing dot diagrams with $|Z_{+}|=d_{+}$, $|Z_{-}|=d_{-}$, and $|Z_{2}|=2d_{2}$. To understand the previously-defined topology of all these spaces one should think of the dots as being allowed to move horizontally in the complex plane. Dots are allowed to cross and dots of a given type are indistinguishable. The ambient spaces $X_{d}^{\prime}$ introduced in §2.5 correspond to letting complex conjugate pairs of dots move arbitrarily in $\mathbb{C}-\mathbb{R}$.

The viewpoint on the $X_{d_{+},d_{-},d_{2}}$ of the previous sections of course remains valid. Thus, as explained before, the closed yellow cones in Figures 3.1 and 3.2 respectively indicate half of $X_{3,0,0}=X_{+++}$ and $X_{1,2,0}=X_{+--}$. Likewise one should think of Figure 3.3 as a picture of the upper half of both $X_{+c}\subset X^{\prime}_{+c}$ and $X_{-c}\subset X^{\prime}_{-c}$, as each full space $X_{\tau c}$ has the form $\frac{1}{2}\mathbb{Z}_{\geq 1}\times\mathbb{R}$.

5.2. Spectral parameters as factorizing real polynomials

Our introduction of coordinates $(\lambda_{1},\dots,\lambda_{d_{1}-1})$ and $(\kappa_{1},\dots,\kappa_{d_{2}})$ on $X^{\prime}_{d_{1},d_{2}}$ in §2.5 involved an ordering convention from §2.2. As is often the case, it is good to have convention-independent coordinates. Accordingly, we form corresponding monic polynomials,

[TABLE]

The distinction between the $+$ points and the $-$ points plays a secondary role, and so it is moreover useful to consider the product,

[TABLE]

Thus a point $(Z_{+},Z_{-},Z_{2})$ of $X^{\prime}_{d}$ determines a point $(c_{2},\dots,c_{d})$ of $\mathbb{R}^{d-1}$. The dual of $(Z_{+},Z_{-},Z_{2})$ is $(-Z_{+},-Z_{-},-Z_{2})$. In the new coordinates, dualizing corresponds to replacing $f(x)$ by $(-1)^{d}f(-x)$ so that $c_{j}$ is replaced by $(-1)^{j}c_{j}$.

Let $Y_{d}\subset\mathbb{R}^{d-1}$ be the image of $X_{d}$ and let $Y_{d_{1},d_{2}}$ be the image of $X_{d_{1},d_{2}}$. The map $X_{d_{1},d_{2}}\rightarrow Y_{d_{1},d_{2}}$ corresponds to forgetting the distinction between the $+$ and $-$ points. It is thus a branched cover of degree $2^{d_{1}}$. Numerics are governed by a row of Pascal’s triangle, as the branched cover $X_{d_{+},d_{-},d_{2}}\rightarrow Y_{d_{1},d_{2}}$ has degree the binomial coefficient $\frac{d_{1}!}{d_{-}!d_{+}!}$. These numerics stay the same if we consider the enlarged spaces $X^{\prime}_{d_{+},d_{-},d_{2}}$ and their images $Y^{\prime}_{d_{1},d_{2}}$.

5.3. The coefficient landscape

Figure 5.2 draws a window on the upper half of the coefficient plane $Y^{\prime}_{3}=\mathbb{R}^{2}$. Also included are the data points from Figures 3.1, 3.2, 3.3, and their duals. The space $Y_{3,0}$ is the subspace where the discriminant $D=-27c_{3}^{2}-4c_{2}^{3}$ is positive or zero. It is filled by blue points coming from the bijection $X_{+++}\rightarrow Y_{3,0}$ and red points from the triple cover $X_{+--}\rightarrow Y_{3,0}$.

For a real number $\kappa$, let $C_{\kappa}$ be the contour curve with equation $D=-4\kappa^{6}$, so that $C_{0}$ is the drawn discriminant locus bounding $Y_{3,0}$. The space $Y_{1,1}$ is a union of curves $C_{\kappa}$ indexed by $\kappa\in\frac{1}{2}\mathbb{Z}_{\geq 1}$. The maps from $X_{-c}$ and $X_{+c}$ to $Y_{1,1}$ are both bijective. The curve $C_{\kappa}$ contains purple points from $X_{-c}$ or green points from $X_{+c}$, according to whether $\kappa$ is an integer or not.

A key feature of Figure 5.2 and also Figure 6.2 is a decomposition of $\mathbb{R}^{d-1}$ into two regions via the discriminant $D$, a left one where $D\geq 0$ and a right one where $D<0$. In the general case, there are $\lceil d/2\rceil$ parts, the images of the $X^{\prime}_{d_{1},d_{2}}$. All the components of $Y_{d}$ are entirely inside one of these parts. The components of $Y_{d}$ in the image of $X^{\prime}_{d_{1},d_{2}}$ all have codimension $d_{2}$.

6. Plancherel measure

In the parts near the origin, where data should be at least nearly complete, there is a noticeable difference between the density of L-points in parameter spaces, as illustrated by Figures 3.1, 3.2, and 3.3, and the density of L-points in coefficient space, as illustrated by Figure 5.2. Here we explain how Plancherel measure gives a clear theoretical framework which in particular explains the visual difference.

6.1. The global meaning of Plancherel measure

Plancherel measure is naturally defined via the group $\operatorname{GL}_{d}(\mathbb{R})$ and plays a large role in many purely local questions. However, we are interested in this paper only in its conjectural role in counting L-functions and so will take a more global viewpoint. For a measurable set $U\subset X_{d}$ and a positive integer $N$, let ${\mathcal{L}}(U,N)$ be the set of L-functions with $\Gamma$-factor in $U$ and conductor $N$. Then the governing principle is that one should have

[TABLE]

for a multiplicative function $\nu_{d}$ on positive integers coming from analogous Plancherel measures for the $p$-adic groups $\operatorname{GL}_{d}(\mathbb{Q}_{p})$. Rigorously, one conjectures an asymptotic equivalence as the cutoff $x$ goes to infinity,

[TABLE]

The general asymptotic equivalence (6.2) can only hold for at most one pair of measures, so (6.2) is in particular a conjectural global characterization of the Plancherel measure $\mu_{d}$. Theorem 1 of [14], with a special case given in (6.6) below, is one of many strong statements in the literature supporting the general truth of (6.2).

As a simple illustration of this formalism, let $d=1$ where $X_{1}=X_{+}\coprod X_{-}$. Each summand on the right has one element and maps bijectively to $Y_{1}=\{0\}=\mathbb{R}^{0}$. The number $\nu_{1}(p^{j})$ is the number of characters $(\mathbb{Z}/p^{j})^{\times}\rightarrow\mathbb{C}^{\times}$ that do not factor through the quotient group $(\mathbb{Z}/p^{j-1})^{\times}$. Thus, e.g., $\nu_{1}(p)=p-2$ for any prime $p$. To give a primitive Dirichlet character $\chi:(\mathbb{Z}/N)^{\times}\rightarrow\mathbb{C}^{\times}$ with $N=\prod_{p}p^{e_{p}}$ is to give an injective character $\chi_{p}:(\mathbb{Z}/p^{e_{p}})^{\times}\rightarrow\mathbb{C}^{\times}$ for each $p$. For $N\geq 2$ the two possible values $1$ and $-1$ arise equally often for $\chi(-1)$. The Plancherel measure $\mu_{1}$ gives mass $1/2$ to both $X_{+}$ and $X_{-}$. So in fact (6.1) holds as an equality for all $U$ and all $N\geq 2$. The behavior at $N=1$ represents a doubling phenomenon which holds for all $d$: half the $\Gamma$-factors are excluded and the others should asymptotically occur at twice their Plancherel density.

6.2. Modified absolute values and near-constant densities

This subsection and the next are essentially translations of presentations in the literature into more intuitive terms. See, e.g., [11] for a more Lie-theoretic presentation. As a preliminary for writing down Plancherel measures on parameter spaces $X_{d_{1},d_{2}}$ in an intuitive way, define

[TABLE]

To help understand Plancherel measures on coefficient spaces $Y_{d}$, restrict to $t\geq 0$ and consider the change of variables $s=t^{2}$, and thus $t=s^{1/2}$ and $dt=\frac{1}{2}s^{-1/2}ds$. Then the differential form $tf(t)dt$ is equal to $s^{1/2}f(s^{1/2})\frac{1}{2}s^{-1/2}ds=\frac{1}{2}f(s^{1/2})ds$.

Figure 6.1 shows that $|t|_{+}$ has a double-zero at $t=0$ while $|0|_{-}=2/\pi$. This distinction is important when comparing with data for large $N$, but in our context of $N=1$ it is negligible, as we will discuss further. More important to us is the fact, obvious from the defining formulas, that the modifications $|t|_{\epsilon}$ each approach $|t|$ exponentially fast as $|t|\rightarrow\infty$. So the corresponding densities in the $s$-variable likewise converge rapidly to $1/2$.

6.3. Plancherel measure and its near-Euclidean nature

For constants $P_{d}$ specified in the next subsection, the Plancherel measure is given on the $(d_{1}+d_{2}-1)$-dimensional part $X_{d_{1},d_{2}}$ of $X_{d}$ by

[TABLE]

Here the $z_{i}$ run over the points in the dot diagram and $\epsilon_{i}$ is $+$, $-$ or [math] according to whether $z_{i}$ is in $Z_{+}$, $Z_{-}$, or $Z_{2}$. The factor $2^{d_{1}}$ in the denominator arises from the previously mentioned fact that the map from $X_{d_{1},d_{2}}$ to its image $Y_{d_{1},d_{2}}$ has degree $2^{d_{1}}$. The factor $2^{d_{2}}$ goes away in (6.4) because each of the $\kappa_{i}$ are allowed to be $\frac{1}{2}\mathbb{Z}_{\geq 1}$, rather than just in $\mathbb{Z}_{\geq 1}$.

Formula (6.3) lets one supplement the vision of moving points in dot diagrams: the points tend to repel each other in the sense that moving any two points further apart contributes to an increase in Plancherel density. At close range, the distinction between the subscripts $+$, $-$, and [math] is important, but at long range it is not.

In our cases in the setting $d=3$, using $\lambda_{1}+\lambda_{2}+\lambda_{3}=0$ in the first two cases, (6.3) becomes these more explicit formulas:

[TABLE]

The first two densities are indicated by light contour plots in the background in Figures 3.1 and 3.2 respectively, where they are near-indistinguishable. While the third measure is supported on a union of vertical lines in Figure 3.3, the background there nonetheless gives the contour plot of $\kappa(4\kappa^{2}+9\lambda^{2})$ on the entire positive quadrant.

Now consider the enlarged parameter spaces $X_{d}^{\prime}$ obtained by allowing each $\kappa_{j}$ to run over $(0,\infty)$ as discussed before. On each piece $X^{\prime}_{d_{1},d_{2}}$ define

[TABLE]

A standard fact about discriminants, explained in [12] for example, then says that the push-forward of $\mu^{\prime}_{d}$ to coefficient space $\mathbb{R}^{d-1}$ is just the Euclidean measure $P_{d}\,dc_{2}\dots dc_{d}$. Keeping in mind that the differences between the three $|t|_{\epsilon}$ decay exponentially as $t$ becomes large, as illustrated by Figure 6.1, and our root distances are almost always large, as illustrated by Figure 5.1, the conclusion is that Plancherel measure on coefficient space is extremely close to a Euclidean measure for our purposes.

6.4. Plancherel constants

To get $P_{d}$, we compare the main theorem of [14], as made explicit by [10], with a much more elementary result from [12].

Laplacian asymptotics. The Laplacian eigenvalue associated to an automorphic L-function with $(d_{1},d_{2})=(d,0)$ and spectral parameters $(\lambda_{1},\dots,\lambda_{d})$ in the conventions of [14] is

[TABLE]

Placing an instance of $|{\mathcal{L}}(U,1)|$ from (6.1) in the notation of [14], let $N(T)$ be the number of such L-functions with conductor $N=1$ and Laplacian eigenvalue $\Lambda\leq T$. The symmetric space $M=K\setminus\operatorname{PGL}_{d}(\mathbb{R})/\operatorname{PGL}_{d}(\mathbb{Z})$ has dimension $m=\sum_{j=2}^{d}j=(d+2)(d-1)/2$. Theorem 1 of [14], specialized to the group $\operatorname{PGL}_{d}(\mathbb{R})$, gives the asymptotic

[TABLE]

Here the volume depends on the ratio of choices of Haar measure for $\operatorname{PGL}_{d}(\mathbb{R})$ and $K$. With the choices described in [14, Section 3], the volume is calculated in [10, Theorem 1.6.1] as

[TABLE]

A Euclidean volume. On the other hand, the part $Y_{d,0}^{r}$ of coefficient space $\mathbb{R}^{d-1}$ coming from $(\lambda_{1},\dots,\lambda_{d})$ with $\lambda_{1}^{2}+\cdots+\lambda_{d}^{2}\leq r^{2}$ is just the part of $Y_{d,0}$ with $c_{2}\geq-r^{2}/2$. For example the upper half of $Y_{3,0}^{\sqrt{2000}}$ is the part of Figure 5.2 beneath drawn discriminant curve $C_{0}$. The Euclidean volume of $Y_{d,0}^{r}$, with respect to the measure $dc_{2}\cdots dc_{d}$, is calculated via a Selberg integral in [12, Props. 1.1 and 1.2] to be

[TABLE]

Note that the right sides of (6.6), (6.7), and (6.8) all have a slightly different form from the right sides in the references. We have made these small changes, using facts like the duplication formula for the $\Gamma$-function, so as to make the simplification (6.9) short.

Comparison. If we replaced $T$ on the right side of (6.6) by $T-1$, the asymptotic would still be true. So, the desired Plancherel constant $P_{d}$ is the ratio of the right sides of (6.6) and (6.8), with $T$ replaced by $r^{2}/2$ in conformity with (6.5):

[TABLE]

The final formula is written so that the factor $\zeta(j)/\pi^{j}$ is rational for even $j$. Special cases of (6.9) are $P_{1}=1$, $P_{2}=\frac{1}{12}$ and the evaluation (1.2) of $P_{3}$ highlighted in the introduction.

6.5. Comparison of data and theory in $d=2$.

In this subsection and the next, we compare three things: computed data, Plancherel measure $\mu_{d}$, and the Euclidean approximation $\mu_{d}^{\prime}$ to Plancherel measure. Here we summarize the well-understood $d=2$ case, using it to calibrate our expectations for our $d=3$ case.

We are guided in the $d=2$ case by Figure 6.2, which is to be compared with Figure 5.2 for the $d=3$ case. The Euclidean approximation $\mu_{2}^{\prime}=\frac{dc_{2}}{12}$ gives mass $50$ to both the intervals $[-600,0]$ and $[0,600]$. But, as indicated by the colored L-points, taken from [15], there are only $19$ L-functions in the negative half, giving the very low percentage $38\%$. For the positive half, write always $k=2\kappa+1$ and thus $\kappa=(k-1)/2$. An L-point at $\kappa\in\frac{1}{2}\mathbb{Z}_{\geq 1}$ arises with multiplicity the dimension of the space of cusp forms $S_{k}(1)$ of weight $k$ on $\operatorname{SL}_{2}(\mathbb{Z})$. For odd $k$, this dimension is always [math]. For $k=12$, $14$, $16$, $18$, $20$, and $22$ the dimensions are $1$, [math], $1$, $1$, $1$, and $1$. Adding $12$ to an even $k$ increases the multiplicity by $1$, for a general approximate formula of $(k-1)/12$. These multiplicities are indicated by the areas of the L-points, with the rightmost drawn L-point coming from $k=48$ and its multiplicity of $4$. The total number of L-points, including multiplicity, works out to $37$, for the still small percentage of $74\%$. So we are interested here in obtaining some understanding of why these percentages are so small.

For $c_{2}>0$, it is helpful to use the function $f(k)=((k-1)/2)^{2}$. Plancherel measure is supported on $f(\mathbb{Z}_{\geq 2})$. By the case of (6.3) with no differentials it gives mass $(k-1)/24$ to $f(k)$. A natural cutoff is $C=f(K+1/2)$ for a positive integer $K$. Then, assuming that $K$ is a multiple of $12$ for the last formula,

[TABLE]

So the low-percentage problem is not at all a failure of $\mu_{2}^{\prime}$ to be a good approximation for $\mu_{2}$. One can rewrite the discrepancy between the theories (6.10), (6.11) and the data (6.12) in the $c_{2}$ variable. One then gets that for $c_{2}>10$ the density

[TABLE]

approximates the density of L-points much better than $\frac{1}{12}$ does. Note also that the doubling phenomenon arises here: at level $N=1$, odd $k$ never occur and so even $k$ occur twice as often as Plancherel measure would predict.

For $c_{2}<0$ there is still a doubling phenomenon: at level $N=1$, dot diagrams of the form $+-$ cannot occur and so $++$ and $--$ asymptotically occur twice as often as Plancherel measure would predict. So both $++$ and $--$ should have asymptotic average density $1/24$. This exact constant comes from the Euclidean measure $\mu_{2}^{\prime}$ and is graphed in Figure 6.2. The density for actual Plancherel measure $\mu$ would visually differ from $1/24$ only in the $c_{2}$ interval $[-2,0]$, by the asymptotic constancy of the right half of Figure 6.1. Five-term refinements for the counting functions with a very small error terms are given in [22, Eqs. 40,41]. Taking only three terms, the corresponding densities written in terms of $b=|c_{2}|$ are

[TABLE]

These functions are plotted in Figure 6.2. Their low values over the plotted region are consistent with there being only $6$ L-functions of type $++$ and $13$ L-functions of type $--$, rather than the $25$ of each that Plancherel measure alone would predict. The slow approach of these functions to their common asymptote is illustrated by the numerical values $24f_{++}(10000)\approx 77\%$ and $24f_{--}(10000)\approx 89\%$.

6.6. Comparison of data and theory in $d=3$

We can now compare the coefficient landscape drawn in Figure 5.2 with expectations from theory. The Plancherel measure of any rectangle of size say $200\mbox{-by-}2000$ is almost exactly $(200)(2000)P_{3}\approx 210$. From the previous subsection, we should not expect to obtain this density in the range of the figure, but we should expect to be closer in the region with $d_{1}=1$ to the right than we are in the region with $d_{1}=3$ to the left. A greater density is indeed visually apparent on the right. For example, the rectangle with $-300\leq c_{2}\leq-100$ and $2000\leq c_{3}\leq 4000$ has $119$ green and purple L-points, giving the percentage $119/210\approx 57\%$. In comparison, the rectangle with $-500\leq c_{2}\leq-300$ and $0\leq c_{3}\leq 1000$ in the totally real region on the left has only $82$ blue and red L-points, which is about $39\%$ of $210$.

We think it is possible that we have found all the L-functions in the two sample rectangles just mentioned. If one goes closer to $(c_{2},c_{3})=(0,0)$, we think it is more likely that we have found all the L-functions. Here the percentages are generally lower, and we interpret that as being from secondary effects, capturing not only that there are no L-points in the L-point-free regions discussed in §4.3, but that L-points close to these regions should be sparse. If one goes further from $(c_{2},c_{3})=(0,0)$ again the experimental density tends to decrease, but now it is clear that our searches have simply missed L-functions.

In interpreting the difference between the blue L-points from $X_{+++}$ and the red L-points from $X_{+--}$ on the totally real side, one needs to keep in mind that there are asymptotically three times as many of the latter as there are of the former. But even with this in mind, the data near the discriminant locus $C_{0}$ suggests that secondary effects for +++ are much stronger than they are for +$$-$$-; this is expected from comparison of the $++$ and the $--$ cases in $d=2$. More data would be very helpful in making a more detailed analysis. Even more helpful would be refined densities with more terms, analogous to (6.13), (6.14), and (6.15).

7. Complements

We conclude by pointing to two promising future directions. We describe them briefly here. Both involve the partially conjectural concept of the Sato-Tate group associated to an L-function.

7.1. Comparison with the $p$-adic Plancherel formula

Another way to examine our collection of L-functions is to compare it with expectations from the spherical $p$-adic Plancherel formula. This formula asymptotically governs the distribution of coefficients $a_{p}\in\mathbb{C}$ for a fixed prime $p$ and varying L-functions. There are uniform formulas for all $d$, but we will be briefer than in the last section and consider only our case $d=3$.

To see some data before bringing in theory, Figure 7.1 plots the $a_{2}$’s and $a_{5}$’s of all the degree 3 conductor 1 L-functions in our collection. Except in a few cases those L-functions are not self-dual, which explains the reflection symmetry across the real axis. The Ramanujan conjecture says that each $a_{p}$ should agree with the trace of an element of the compact group $\operatorname{SU}_{3}$. The conjecture holds here, as expected, because the $a_{p}$ indeed lie in the curvilinear triangle $T$ of traces. Writing the generic element in the complex plane as $z=x+iy$, the boundary of $T$ is given by the vanishing of

[TABLE]

For a fixed non-self-dual L-function and varying $p$, the $a_{p}$ are conjectured to be equidistributed in $T$ according to the push-forward of the Haar measure on $\operatorname{SU}_{3}$. The density of this Sato-Tate measure $\mu_{\infty}=f_{\infty}(x,y)dxdy$ works out to $f_{\infty}(x,y)=\sqrt{f(x,y)}/(2\pi^{2})$.

The $p$-adic Plancherel measures $\mu_{p}=f_{p}(x,y)dxdy$ fit together to form a family indexed by $p\in[1,\infty]$, with $p=\infty$ giving the Sato-Tate measure just discussed. The density $f_{1}(x,y)$ is constant. The formula for general $p$ is too complicated to give here, but for $p>1$ all the $f_{p}(x,y)$ vanish on the boundary, and the unique maximum is at $(0,0)$ and increases with $p$. A precise statement which captures this phenomenon of increasing concentration in the middle is

[TABLE]

We obtained all these statements by specializing the general definition of $p$-adic Plancherel measure [16] to $\operatorname{SU}_{3}$ and then carrying out the requisite multivariate calculus.

Figure 7.1 shares qualitative features with the behavior predicted by $p$-adic Plancherel measures. For example, the distribution of the $a_{2}$’s is hard to visually distinguish from the uniform distribution, but the distribution of the $a_{5}$’s already shows marked repulsion from the boundary. However, quantitatively, experimental moments are far off from theoretical moments:

[TABLE]

We again expect that this discrepancy is explained by as yet unknown secondary terms.

7.2. Group-theoretical framework

To make our concrete points in the next two subsections cleanly, we first bring in more formalism. To understand the full set ${\mathcal{L}}$ of all automorphic L-functions it is clarifying to use the conjectural Langlands group $G$ and two of its quotients:

[TABLE]

Here $G$ is a very large compact group containing Frobenius classes $\mbox{Fr}_{p}$ for all primes $p$. The set of irreducible representations of $G$ is identified with ${\mathcal{L}}$ by requiring that a representation $\rho$ corresponding to an L-function $L(s)=\prod_{p}f_{p}(p^{-s})^{-1}$ satisfy $f_{p}(x)=\det(1-\rho(\mbox{Fr}_{p})x)$ for all primes $p$ not dividing the conductor. The representations that factor through $G^{\rm alg}$ correspond to L-functions in ${\mathcal{L}}^{\rm alg}$, therefore L-functions for which the $\lambda_{j}$ are all [math]. The representations that factor through $G^{\rm fin}$ correspond to L-functions for which the $\kappa_{k}$ are also all [math]. Conjecturally, $G^{\rm alg}$ has an independent definition as a compact version of the absolute motivic Galois group of $\mathbb{Q}$. Conjecturally, $G^{\rm fin}=\mbox{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. The Sato-Tate group of an L-function $L$ is the image of the corresponding representation $\rho$; it is a compact subgroup of $\operatorname{GL}_{d}(\mathbb{C})$, well-defined up to conjugation. In general Frobenius elements are conjectured to be equidistributed in the conjugacy classes of the Sato-Tate group, as mentioned in the previous subsection for the Sato-Tate group $\operatorname{SU}_{3}$.

If the Sato-Tate group of an L-function has $c$ components, then the component group conjecturally comes from a Galois field of degree $c$. Any prime dividing the discriminant of this field divides the conductor of the L-function. So requiring $N=1$ forces Sato-Tate groups to be connected. Also the abelianizations of the three groups (7.2) are expected to coincide with the abelianization $\hat{\mathbb{Z}}^{\times}$ of $\mbox{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. So, at $N=1$, tori are also ruled out as possible Sato-Tate groups.

For $d=3$ and $N=1$ there are only two possibilities. The Sato-Tate group is $\operatorname{SO}_{3}$ in the special case that the L-function is self-dual and $\operatorname{SU}_{3}$ otherwise. This simple dichotomy allowed us to ignore the group-theoretical framework in previous sections. But when one allows general $N$, there are infinitely many possible $G$, with the finite groups arising in §3.4 illustrating some possibilities. To regain some simplicity in discussing the landscapes for $(d,N)$, one can focus on generic L-functions. By definition, these are L-functions whose Sato-Tate groups contain $\operatorname{SU}_{d}$, with $d$ being the degree. One expects that for any fixed $d$, $100\%$ of L-functions are generic asymptotically, for say a ball centered at the origin in coefficient space $\mathbb{R}^{d-1}$ of increasing radius.

7.3. Generic algebraic L-functions are hard to find for $d\geq 3$

We now support a main theme of the introduction, that algebraic L-functions are relatively rare for degrees $d\geq 3$. The classical degrees do not point in the right direction at all: in degree $1$, all L-functions are algebraic. For $d=2$, an L-function is algebraic if and only if $c_{2}\geq 0$ as illustrated by Figure 6.2; so Plancherel measure says in a strong sense that half of the L-functions are algebraic and half are not.

The set $Y_{3}^{\rm alg}$ is just the set of points $(c_{2},c_{3})=(\kappa^{2},0)$ for $\kappa\in\mathbb{Z}_{\geq 0}$. We impose genericity for the rest of this subsection, thereby excluding the parameter $\kappa=0$. Computations with the cohomology of $\operatorname{SL}_{3}(\mathbb{Z})$ in [1] say that there are no generic L-functions in ${\mathcal{L}}^{\rm alg}_{3,1}$ with $N=1$ and $\kappa<120$. To see some algebraic L-functions, we increase the conductor, like we did in §3.4. At $\kappa=1$, cohomological computations with congruence subgroups in [24] show the existence of dual pairs of generic L-functions for the conductors $89$, $106$, $116$, $128$, $160$, $205$, $212$, and $221$. An L-function with L-point $(\kappa^{2},0)$ for $\kappa\in\mathbb{Z}_{\geq 1}$ should come from a motive with nonvanishing Hodge numbers $h^{2\kappa,0}=h^{\kappa,\kappa}=h^{0,2\kappa}=1$. For $\kappa=1$, a one-parameter family of such motives was found in the cohomology of surfaces in [25]. A dual pair of members of this family numerically matches the dual pair of automorphic L-functions with conductor $128$. For $\kappa>1$ there is a qualitative difference, as Griffiths transversality prevents motives from moving in families, and so algebraic geometry can only produce them one dual pair at a time.

In summary, generic algebraic L-functions in degree $d=3$ are known only for $\kappa=1$. For $d\geq 4$, the only examples we know of come from families of threefolds in [21, 5], with nonvanishing Hodge numbers $h^{3,0}=h^{2,1}=h^{1,2}=h^{0,3}=1$; the smallest conductor seems to be $2^{9}3^{9}$, so matching automorphic calculations are out of reach.

7.4. Comparison in the self-dual setting

For a more subtle comparison, consider now only L-functions in ${\mathcal{L}}$ with real coefficients. According to the group-theoretic framework, they come in two types, orthogonal and symplectic, the latter arising only for $d$ odd. In the spirit of the Frobenius-Schur indicator, we use $\tau\in\{+,-\}$ to distinguish them, with $+$ for orthogonal and $-$ for symplectic.

The theoretical context set up in this paper has direct analogs in both of these two parallel settings. Thus there are sets of L-functions ${\mathcal{L}}^{\tau}_{d,N}$ mapping to disconnected parameter spaces $X^{\tau}_{d}$ of possible $\Gamma$-factors which in turn map to coefficient spaces $Y^{\tau}_{d}\subset\mathbb{R}^{r}$. Here $r=\lfloor d/2\rfloor$ arises as the dimension of a maximal torus in the compact group $\operatorname{Sp}_{d}$ or $\operatorname{O}_{d}$. There is again a Plancherel measure $\mu_{\tau}$ on $Y_{d}^{\tau}$ and again a canonical Euclidean approximation $\mu^{\prime}_{\tau}$ on $\mathbb{R}^{r}$. Again $\mu_{\tau}$ should govern the distribution of L-points. A difference is that $Y^{\tau}_{d}$ contains components of all the possible dimensions [math], …, $r$. So Figure 6.2 for $\operatorname{SU}_{2}=\operatorname{Sp}_{2}$ is a good guide. Like in the $d=2$ case, the isolated points are algebraic and each has positive Plancherel measure. Another difference is that algebraic geometry provides many large families of source motives for the algebraic points of many of the larger-dimensional components. For example, the $g(g+1)/2$-dimensional family of dimension $g$ abelian varieties maps to the algebraic point of a $\lfloor g/2\rfloor$-dimensional component of $Y_{2g}^{-}$.

As mentioned in the introduction, some transcendental L-functions representing the symplectic case with $(d,N)=(4,1)$ are already in the LMFDB. The $d=4$ cases with somewhat larger conductors should be within computational reach.