A note on the second moment of automorphic L-functions

TL;DR

This paper derives a formula for the second moment of automorphic L-functions associated with primitive Hecke eigenforms of weight 2, connecting it to recent reciprocity results in the field.

Contribution

It provides a new explicit formula for the twisted harmonic second moment of these L-functions, extending understanding of their mean values and reciprocity properties.

Findings

Derived the formula for the twisted harmonic second moment.

Connected the results to recent reciprocity formulas.

Enhanced understanding of automorphic L-function moments.

Abstract

We obtain the formula for the twisted harmonic second moment of the $L$-functions associated with primitive Hecke eigenforms of weight 2. A consequence of our mean value theorem is reminiscent of recent results of Conrey and Young on the reciprocity formula for the twisted second moment of Dirichlet $L$-functions.