We study zeros of the L-functions L(f, s) of primitive weight two forms of level q. Our main result is that, on average over forms f of level q prime, the order of the L-functions at the central critical point s = 1 2 is absolutely bounded. On the Birch and Swinnerton-Dyer conjecture, this provides an upper bound for the rank of the Jacobian J0(q) of the modular curve X0(q), which is of the sam...
