Moments of $L$-functions and the Liouville–Green method

Abstract

We show that for at least 20% of primitive forms of level 1 and weight $4k\to \infty ,k\in \mathbb{N}$, the associated $L$-function at the central point is no less than (log $k{)}^{–2}$. The key ingredients of our proof are the Kuznetsov convolution formula and the Liouville–Green method.