Non-vanishing of high derivatives of automorphic L-functions at the centerof the critical strip

We prove non-vanishing results for the central value of high derivatives of the complete L-function Lambda(f,s) attached to primitive forms of weight 2 and prime level q. For fixed k greater than or equal to 0 the proportion of primitive forms f such that Lambda((k))(f, 1/2) not equal 0 is greater t han or equal to p(k) + o(1) with p(k) > 0 and p(k) = 1/2 + O(k(-2)), as the level q goes to infinity. This result is (asymptotically in k) optimal and analogous to a result of Conrey on the zeros of high derivatives of Rieman n's xi function lying on the critical line. As an application we obtain new strong unconditional bounds for the average order of vanishing of the form s f (i.e. the analytic rank of the Jacobian variety J(0)(q)).