On the vanishing of twists of L-functions of elliptic curves Chantal ...
On the vanishing of twists of L-functions of elliptic curves Chantal ...
On the vanishing of twists of L-functions of elliptic curves Chantal ...
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When N → ∞R(N) → N 1/4 G 2 (1/2)where G is <strong>the</strong> Barnes G-function.Moments <strong>of</strong> L E (1, χ):M E (s, X) = 1N(X)∑f χ ≤X|L E (1, χ)| s .From <strong>the</strong> correspondance <strong>of</strong> Keating and Snaithwe conjecture that|L E (1, χ)| ↔ |P A (1)|,M E (s, X) = a E (s)M U (s, N)for <strong>the</strong> arithmetical factor a E (s) andN ∼ 2 log X(equating <strong>the</strong> densities <strong>of</strong> zeroes at a fixedheight).Probability distribution for small x:∫P E (X, x) = 12πiM E(s, X)x −s−1 ds(c)∼ a E (−1)R(N)13