We examine the number of vanishings of quadratic twists of the L-function associated to an elliptic curve. Applying a conjecture for the full asymptotics of the moments of critical L-values we obtain a conjecture for the first two terms in the ratio of the number of vanishings of twists sorted according to arithmetic progressions.
Conrey, J. Brian; Pocharel, Atul; Rubinstein, Michael O.; and Watkins, Mark, "Secondary Terms in the Number of Vanishings of Quadratic Twists of Elliptic Curve L-Functions" (2006). Mathematics - Faculty Scholarship. 107.
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