Upper bounds for the lowest first zero in families of cuspidal newforms

Abstract

Assuming the Generalized Riemann Hypothesis, the non-trivial zeros of L-functions lie on the critical line with the real part 1/2. We find an upper bound of the lowest first zero in families of even cuspidal newforms of prime level tending to infinity. We obtain explicit bounds using the n-level densities and results towards the Katz-Sarnak density conjecture. We prove that as the level tends to infinity, there is at least one form with a normalized zero within 0.218503 of the average spacing. We also obtain the first-ever bounds on the percentage of forms in these families with a fixed number of zeros within a small distance near the central point.