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We study the 1-level density of low-lying zeros of quadratic Dirichlet  L -functions by applying the  L -functions Ratios Conjecture. We observe a transition in the main term as was predicted by the Katz–Sarnak heuristic as well as in the lower-order terms when the support of the Fourier transform of the corresponding test function reaches the point 1. Our results are consistent with those obtained in previous work under GRH and are furthermore analogous to results of Rudnick in the function field case.

Low-lying zeros of quadratic Dirichlet L -functions: A transition in the ratios conjecture