The second moment of quadratic twists of modular $L$-functions

TL;DR

This paper investigates the second moment of quadratic twists of modular L-functions, providing unconditional lower bounds and, under GRH, proving the conjectured asymptotic formula for their central values.

Contribution

It establishes the asymptotic behavior of the second moment of quadratic twists of modular L-functions, confirming conjectures under GRH and providing unconditional bounds.

Findings

Unconditional lower bounds match the conjectured asymptotic.

Under GRH, the asymptotic formula for the second moment is proven.

Results advance understanding of the distribution of quadratic twists' central values.

Abstract

We study the second moment of the central values of quadratic twists of a modular $L$-function. Unconditionally, we obtain a lower bound which matches the conjectured asymptotic formula, while on GRH we prove the asymptotic formula itself.