The second moment of Dirichlet twists of Hecke $L$-functions

TL;DR

This paper studies the average behavior of the squared central values of Hecke L-functions twisted by primitive Dirichlet characters, extending previous results to almost all moduli q, beyond those with few prime factors.

Contribution

The authors extend Stefanicki's asymptotic results for the second moment of Dirichlet twists of Hecke L-functions to almost all large moduli q, removing previous restrictions.

Findings

Asymptotic formula valid for almost all q

Extension beyond moduli with few prime factors

Broader understanding of L-function value distribution

Abstract

Fix a Hecke cusp form $f$, and consider the $L$-function of $f$ twisted by a primitive Dirichlet character. As we range over all primitive characters of a large modulus $q$, what is the average behavior of the square of the central value of this $L$-function? Stefanicki proved an asymptotic valid only for $q$ having very few prime factors, and we extend this to almost all $q$.