Random Matrix Theory and the Attraction of Zeros of L-Functions From the Central Point

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