On generalized Li criterion for a certain class of $L-$functions

Abstract

We define generalized Li coefficients, called $\tau-$Li coefficients for a very broad class $\mathcal{S}^{\sharp \flat }(\sigma_0, \sigma_1)$ of $L-$functions that contains the Selberg class, the class of all automorphic $L-$functions and the Rankin-Selberg $L-$functions, as well as products of suitable shifts of those functions. We prove the generalized Li criterion for zero-free regions of functions belonging to the class $\mathcal{S}^{\sharp \flat }(\sigma_0, \sigma_1)$, derive an arithmetic formula for the computation of $\tau-$Li coefficients and conduct numerical investigation of $\tau-$Li coefficients for a certain product of shifts of the Riemann zeta function.