On the asymptotic criterion for the zero-free regions of certain $L$-functions

Abstract

We investigate relations between zero-free regions of certain $L$-functions and the asymptotic behavior of corresponding generalized Li coefficients. Precisely, we prove that violation of the $\tau/2$-generalized Riemann hypothesis implies oscillations of corresponding $\tau$-Li coefficients with exponentially growing amplitudes. Results are obtained for class $\shfs$ that contains the Selberg class, the class of all automorphic $L$-functions, the Rankin--Selberg $L$-functions, and products of suitable shifts of the mentioned functions.