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

Abstract

In this note, we prove that given a Hecke-Maass cusp form $f$ for $S{L}_2(\mathbb{Z})$ and a sufficiently large integer $q=q_1{q}_2$ with $q_j\asymp \sqrt{q}$ being prime numbers for $j=1,2$, there exists a primitive Dirichlet character $\ddot{I}‡$ of conductor $q$ such that $L\left(\frac{1}{2},f\otimes \ddot{I}‡\right)L\left(\frac{1}{2},\ddot{I}‡\right)\ne 0$. To prove this, we establish asymptotic formulas of $L\left(\frac{1}{2},f\otimes \ddot{I}‡\right)L\left(\frac{1}{2},\ddot{I}‡\right)$ over the family of even primitive Dirichlet characters $\ddot{I}‡$ of conductor $q$ for more general $q$.

19 pages. Comments are welcome