The order of vanishing of L-functions at the center of the critical strip /

In this work we investigate the order of vanishing of L( s, chi) and L(s, f) (resp. rchi and rf) at the centre of the critical strip, where chi is a Dirichlet character and f is a newform of weight 2 and level N (f ∫ ∈ S2(N)^new).We show that on average rf or r chi are not large. Precisely: (1) We generalise Murty's results on the analytic rank of J0(N) to arbitrary levels N: under the RH for all L(s, f) then lim sup N→∞ f∫S2(N)new rf dim S2(N) ≤ 3/2. (2) We prove that the triangle function max(1 - |u|, 0) is the best function when using the explicit formulas the way Brumer and Murty did. (3) We compute an explicit bound for the second moment of the harmonic rank of J0(N). We prove that under the RH for L( s, f) and the Linnik-Selberg conjecture we have the bound f∫S2(N)new r2f 4π〈f,f〉 ≤ 31/12+o(1) as N → ∞. (4) We show on higher moments that f∫S2(N)new rkf ≪ dim S2(N), subject to the RH. (5) We improve on the constant 32 above to 1 when N is prime, thus proving that 50% of newforms with root number -1 have a non-vanishing L-function at s = 1 (conditional on the RH for L( s, f) again). (6) We show that assuming the RH for L(s, f) there is, for each N sufficiently large, a newform in S2(N)new so that L(1+it0,f) = 0 with |t0|≤4/log N