Date of Thesis

5-7-2015

Thesis Type

Honors Thesis

Degree Type

Bachelor of Science

First Advisor

Nathan Ryan

Abstract

We explore the attraction of zeros near the central point of L-functions associated with elliptic curves and modular forms. Specifically, we consider families of twists of elliptic curves, the family of weight 2 modular forms, and the family of level 1 modular forms. We observe experimentally an attraction of the zeros near the central point, and that the attraction decreases with the rank r of the L-function. However, for each set of L-functions of rank r within a particular family we observe a statistically significant increase in the attraction as the conductors of the L-functions increase. This indicates a correspondence with the random matrix theory result about the vanishing of the distance between eigenangles near 1 as the size of the matrix increases, but also that this correspondence only exists in the limit, since we observe less attraction otherwise. Additionally, we begin preliminary investigation on a new statistic, the relationship between the value of the first zero above the central point and the value of the L-function at s = 1/2.

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