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We relate the study of Landau-Siegel zeros to the ranks of Jacobians $J_0(q)$of modular curves for large primes $q$. By a conjecture of Brumer-Murty, therank should be equal to half of the dimension. Equivalently, almost allnewforms of weight two and level $q$ have analytic rank $\leq 1$. We show thateither Landau-Siegel zeros do not exist, or that almost all such newforms haveanalytic rank $\leq 2$. In particular, almost all odd newforms have analyticrank equal to one. Additionally, for a sparse set of primes $q$ we show therank of $J_0(q)$ is asymptotically equal to the rank predicted by theBrumer-Murty conjecture.

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