Nonvanishing of L -functions for GL n A Q

Abstract

In this work, we establish new nonvanishing results for automorphic $L$-functions on $\mathrm{GL}\left(n,{\mathbf{A}}_Q\right)$. In particular, we show that, given a cuspidal automorphic form $\pi$ on $\mathrm{GL}\left(3,{\mathbf{A}}_Q\right)$ and an arbitrary point $s_0\in \mathbf{C}$, there exist infinitely many Dirichlet characters $\chi$ with prescribed ramification such that the twisted completed $L$-functions do not vanish at $s_0$: $\Lambda \left(s_0,\pi \otimes \chi \right)\ne 0$.