Explicit upper bound for the (analytic) rank of J(0)(q)

We refine the techniques of our previous paper [KM1] to prove that the aver age order of vanishing of L-functions of primitive automorphic forms of wei ght 2 and prime level q satisfies Sigma (f is an element of S2(q)*) ord(s=1/2) L(f,s) less than or equal to C \S-2(q)*\ with C = 6.5, for all q large enough. On the Birch and Swinnerton-Dyer conj ecture, this implies rank Jo(q) < C dim Jo(q) for q prime large enough.