On extreme values of quadratic twists of Dirichlet-type $L$-functions

TL;DR

This paper demonstrates that quadratic twists of Dirichlet-type L-functions can attain arbitrarily large values at the critical point, extending recent results from cusp form L-functions to a broader class of functions.

Contribution

It establishes large value results for twists of Dirichlet-type L-functions, generalizing previous work on cusp forms to these specific L-functions.

Findings

Many fundamental discriminants yield large L-values at the critical point.

The number of such discriminants grows roughly as X^{1- ext{epsilon}}.

Results hold for a broad class of Dirichlet characters of prime conductor q.

Abstract

In a recent work arXiv:2004.14450, it has been shown that $L$-functions associated with arbitrary non-zero cusp forms take large values at the central critical point. The goal of this note is to derive analogous results for twists of Dirichlet-type functions. More precisely, for an odd integer $q >1$, let $F$ be a non-zero $\mathbb{C}$-linear combination of primitive, complex, even Dirichlet characters of conductor $q$. We show that for any $\epsilon>0$ and sufficiently large $X$, there are $\gg X^{1-\epsilon}$ fundamental discriminants $8d$ with $X < d \leq 2X$ and ${(d, 2q)=1}$ such that ${|L(1/2, F \otimes \chi_{8d})| }$ is large.