Analytic ranks of automorphic L-functions and Landau-Siegel zeros

We relate the study of Landau-Siegel zeros to the ranks of Jacobians J0(q) of modular curves for large primes q. By a conjecture of Brumer-Murty, the rank should be equal to half of the dimension. Equivalently, almost all newforms of weight two and level q have analytic rank ≤ 1. We show that either Landau-Siegel zeros do not exist, or that, for wide ranges of q, almost all such newforms have analytic rank ≤ 2. In particular, in wide ranges, almost all odd newforms have analytic rank equal to one. Additionally, for a sparse set of primes q in a wide range we show the rank of J0(q) is asymptotically equal to the rank predicted by the Brumer-Murty conjecture.