Non vanishing of products of twisted $\mathrm{GL}(3)$ $L$-functions

TL;DR

This paper demonstrates that under certain conditions on Fourier coefficients, there are infinitely many Dirichlet characters for which the product of twisted $ ext{GL}(3)$ and Dirichlet $L$-functions at the central point is non-zero, using a first moment calculation.

Contribution

It establishes a non-vanishing result for products of twisted $ ext{GL}(3)$ and Dirichlet $L$-functions, advancing understanding of their simultaneous non-vanishing properties.

Findings

Proves existence of infinitely many non-vanishing $L$-function products.

Computes the first twisted moment over a set of conductors.

Shows non-correlation condition on Fourier coefficients implies non-vanishing.

Abstract

In this paper, we prove that if the Fourier coefficients of a $\mathrm{SL}(3,\mathbb{Z})$ Hecke--Maa\ss\ cusp form $\pi$ are not too correlated with additive characters, then there exists infinitely many Dirichlet characters such that \begin{align*} L\left(\frac{1}{2},\pi\otimes\chi\right)L\left(\frac{1}{2},\chi\right)\neq 0. \end{align*} To prove this result, we compute the first twisted moment of these $L$ function averaged over a well chosen set of conductors.