#### Date of Thesis

2015

#### Thesis Type

Honors Thesis

#### Degree Type

Bachelor of Science

#### Major

Mathematics

#### First Advisor

Nathan Ryan

#### Keywords

Random matrix theory, l-functions, Modular forms, Elliptic curves

#### 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.

#### Recommended Citation

Harris, Katherine Elizabeth, "Random Matrix Theory and the Attraction of Zeros of L-Functions From the Central Point" (2015). *Honors Theses*. 321.

https://digitalcommons.bucknell.edu/honors_theses/321