Simultaneous nonvanishing of Dirichlet $L$-functions and twists of Hecke-Maass L-functions

Abstract

We prove that given a Hecke-Maass form $f$ for $\text{SL}(2, \mathbb{Z})$ and a sufficiently large prime $q$, there exists a primitive Dirichlet character $\chi$ of conductor $q$ such that the $L$-values $L(\tfrac{1}{2}, f \otimes \chi)$ and $L(\tfrac{1}{2}, \chi)$ do not vanish. We expect the same method to work for any large integer $q$