A note on the standard zero-free region for $L$-functions

TL;DR

This paper establishes a standard zero-free region for a broad class of $L$-functions, including Rankin--Selberg $L$-functions, based on properties of their logarithmic coefficients.

Contribution

It introduces a zero-free region for $L$-functions with nonnegative real part coefficients, extending known results to a wider class of functions.

Findings

Zero-free region established for a general class of $L$-functions.

Includes Rankin--Selberg $L$-functions for automorphic representations.

Applicable to $L$-functions with nonnegative real part coefficients.

Abstract

In this short note, we establish a standard zero-free region for a general class of $L$-functions for which their logarithms have coefficients with nonnegative real parts, which includes the Rankin--Selberg $L$-functions for unitary cuspidal automorphic representations.