Non-Vanishing of Cubic Twists of $G{L}_n(\mathbb{Q})$ $L$-functions

Abstract

Let $\pi$ be an irreducible, cuspidal automorphic representation of $G{L}_n({\mathbb{A}}_{\mathbb{Q}})$ ($n\ge 3$), which is tempered only for $n=3$. Let $s$ be a complex number such that $\Re (s)\notin \left[1/n,1-1/n\right]$ if $n\ne 4$; $\Re (s)\notin \left[1/5,4/5\right]$ if $n=4$, then we show that there are infinitely many primitive cubic Dirichlet characters $\chi$ such that $L(s,\pi \times \chi )\ne 0$. Similar results were previously known only for primitive Dirichlet characters without any restriction on the order and quadratic Dirichlet characters.

Comments are welcome!