Simultaneous non vanishing of $GL(3)$ $L$-functions

TL;DR

This paper investigates the simultaneous non-vanishing of $GL(3)$ $L$-functions at the central point by computing a first moment involving Dirichlet and twisted self-dual $L$-functions, utilizing symmetric square $L$-function properties.

Contribution

It introduces a method to analyze the simultaneous non-vanishing of $GL(3)$ $L$-functions at the central point through first moment computations.

Findings

Established a first moment formula for the product of Dirichlet and $GL(3)$ $L$-functions.

Provided insights into the conditions for simultaneous non-vanishing at the central point.

Utilized properties of symmetric square $L$-functions to facilitate the analysis.

Abstract

The main objective of this article is to compute a first moment for product of Dirichlet and twisted self-dual $GL(3)$ $L$-functions. We discuss the possible simultaneous non vanishing at the central point. We use properties of symmetric squares $L$-functions.