Repulsion of zeros close to $s=1/2$ for L-functions

Abstract

In this paper we present results of several experiments in which we model the repulsion of low-lying zeros of L-functions using random matrix theory. Previous work has typically focused on the twists of L-functions associated to elliptic curves and on families that can be modeled by unitary and orthogonal matrices. We consider families of L-function of modular forms of weight greater than 2 and we consider families that can be modeled by symplectic matrices. Additionally, we explore a model for low-lying zeros of twists that incorporates a discretization arising from the Kohnen--Zagier theorem. Overall, our numeric evidence supports the expectation that the repulsion of zeros decreases as the conductor of the twist increases. Surprisingly, though, it appears that using the discretization that arises from the Kohnen--Zagier theorem does not model the data better than if the discretization is not used for forms of weight 4 or above.