Super-positivity of a family of L-functions in the level aspect

TL;DR

This paper demonstrates that a significant proportion of automorphic L-functions in the level aspect exhibit super-positivity and lack real zeros beyond the central point, advancing understanding of their zero distribution and positivity properties.

Contribution

It proves that at least 12% of certain automorphic L-functions have the super-positivity property and at least 49% have no real zeros off the central point.

Findings

At least 12% of L-functions have super-positivity.

At least 49% of L-functions lack real zeros outside the central point.

The results apply to L-functions associated with Hecke cusp forms of large prime level.

Abstract

An automorphic self dual L-function has the super-positivity property if all derivatives of the completed L-function at the central point $s=1/2$ are non-negative and all derivatives at a real point $s > 1/2$ are positive. In this paper we prove that at least 12% of L-functions associated to Hecke basis cusp forms of weight $2$ and large prime level $q$ have the super-positivity property. It is also shown that at least 49% of such L-functions have no real zeros on $ \Re(s) > 0$ except possibly at $s = 1/2.$