Non-vanishing of high derivatives of automorphic L-functions at the center of the critical strip
نویسنده
چکیده
We prove non-vanishing results for the central value of high derivatives of the complete L-function Λ(f, s) attached to primitive forms of weight 2 and prime level q. For fixed k ≥ 0 the proportion of primitive forms f such that Λ(f, 1/2) 6= 0 is ≥ pk +o(1) with pk > 0 and pk = 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 Riemann’s ξ function lying on the critical line. As an application we obtain new strong unconditional bounds for the average order of vanishing of the forms f (i.e. the analytic rank of the Jacobian variety J0(q)).
منابع مشابه
Twisted Moments of Automorphic L-functions
We study the moments of the symmetric power L-functions of primitive forms at the edge of the critical strip twisted by the square of the value of the standard L-function at the center of the critical strip. We give a precise expansion of the moments as the order goes to infinity. CONTENTS
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