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

TL;DR

This paper proves the existence of infinitely many primitive cubic Dirichlet characters for which the twisted $L$-functions of certain automorphic representations do not vanish at specific complex points, extending previous results.

Contribution

It establishes non-vanishing results for cubic twists of $GL_n(Q)$ $L$-functions, a case not previously covered in the literature.

Findings

Infinitely many primitive cubic Dirichlet characters with non-zero twisted $L$-values.

Results extend non-vanishing to cubic characters, beyond quadratic and general primitive characters.

Applicable for automorphic representations of $GL_n$ with $n eq 4$ under certain real part conditions.

Abstract

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