Towards Keating-Snaith's conjecture for cubic Hecke $L$-functions over the Eisenstein field

TL;DR

This paper provides a conditional lower bound supporting Keating-Snaith's conjecture for the distribution of central values of cubic Hecke L-functions over the Eisenstein field, extending previous zero-density estimates under GRH.

Contribution

It introduces new twisted estimates of zero densities for cubic Hecke L-functions, advancing understanding of their value distribution in a thin family.

Findings

Conditional lower bound towards Keating-Snaith conjecture

Extension of zero-density estimates under GRH

Supports log-normal distribution hypothesis for L-values

Abstract

A famous conjecture of Keating and Snaith asserts that central values of $L$-functions in a given family admit a log-normal distribution with a prescribed mean and variance depending on the symmetry type of the family. Based on a recent work of Radziwill and Soundararajan, we obtain a conditional lower bound towards Keating-Snaith's conjecture for a "thin" family of cubic Hecke $L$-functions over the Eisenstein field. A key new input is certain twisted estimates of the 1-level density of zeros of cubic Hecke $L$-functions, extending the previous work of David and G\"{u}lo\u{g}lu, under the Generalised Riemann Hypothesis.