A note on simultaneous nonvanishing of Dirichlet $L$-functions and twists of Hecke-Maass $L$-functions

TL;DR

This paper proves the simultaneous nonvanishing of certain twisted $L$-functions associated with Hecke-Maass forms and Dirichlet characters, using asymptotic formulas over families of characters for large conductors.

Contribution

It establishes new asymptotic formulas for products of $L$-functions over families of Dirichlet characters, demonstrating nonvanishing at the critical point for large conductors.

Findings

Existence of primitive characters with nonvanishing $L$-values

Asymptotic formulas for $L$-function products over character families

Nonvanishing results for large conductors

Abstract

In this note, we prove that given a Hecke-Maass cusp form $f$ for $SL_2(\mathbb{Z})$ and a sufficiently large integer $q=q_1q_2$ with $q_j\asymp \sqrt{q}$ being prime numbers for $j=1,2$, there exists a primitive Dirichlet character $\chi$ of conductor $q$ such that $L\left(\frac{1}{2},f\otimes \chi\right)L\left(\frac{1}{2},\chi\right)\neq 0$. To prove this, we establish asymptotic formulas of $L\left(\frac{1}{2},f\otimes \chi\right)L\left(\frac{1}{2},\chi\right)$ over the family of even primitive Dirichlet characters $\chi$ of conductor $q$ for more general $q$.