On the complex moments of symmetric power $L$-functions at $s = 1$

Abstract

The study of the distribution values of L-functions at the point s = 1 is a classical topic in analytic number theory which goes back to Chowla and Erdös. For some time, this question has been limited to the case of Euler products of degree 1, and it is only recently that the case of L-functions of higher degree was investigated, firstly by Luo and then by Royer et al., namely, for the standard and the symmetric square L-functions of modular forms. In this paper we propose a conceptual approach to analyzing the distribution of values at s = 1 of automorphic L-functions for appropriate families of automorphic forms. We illustrate this by computing arbitrary complex moments of arbitrary symmetric power L-functions of holomorphic forms of large (prime) level at s = 1 (assuming that the corresponding L-functions are automorphic, as is predicted by the Langlands functoriality conjectures and effectively proved for the symmetric powers up to 4). In addition, we provide a natural probabilistic interpretation of these computations. Our approach should generalize to quite arbitrary L-functions of appropriate families of automorphic forms.