On the value-distribution of the logarithms of symmetric square L-functions in the level aspect

TL;DR

This paper studies the distribution of logarithmic values of symmetric square L-functions for certain modular forms, revealing their connection to the Sato-Tate measure and extending to symmetric power L-functions.

Contribution

It establishes a link between the value distribution of symmetric square L-functions and the Sato-Tate measure, providing new insights into their average behavior and extending the analysis to symmetric power L-functions.

Findings

Average values expressed as integrals with a density related to Sato-Tate measure

Distribution results for symmetric square L-functions at real s>1/2

Discussion of symmetric power L-functions case

Abstract

We consider the value distribution of logarithms of symmetric square L-functions associated with newforms of even weight and prime power level at real s> 1/2. We prove that certain averages of those values can be written as integrals involving a density function which is related with the Sato-Tate measure. Moreover, we discuss the case of symmetric power L-functions.