# 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.

## Key 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.