Geodesic
Weighted Distribution of Low-lying Zeros of GL(2) $L$-functions
Canadian journal of mathematics, Tome 71 (2019) no. 1, pp. 153-182

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DOI

We show that if the zeros of an automorphic $L$-function are weighted by the central value of the $L$-function or a quadratic imaginary base change, then for certain families of holomorphic GL(2) newforms, it has the effect of changing the distribution type of low-lying zeros from orthogonal to symplectic, for test functions whose Fourier transforms have sufficiently restricted support. However, if the $L$-value is twisted by a nontrivial quadratic character, the distribution type remains orthogonal. The proofs involve two vertical equidistribution results for Hecke eigenvalues weighted by central twisted $L$-values. One of these is due to Feigon and Whitehouse, and the other is new and involves an asymmetric probability measure that has not appeared in previous equidistribution results for GL(2).
DOI : 10.4153/CJM-2018-013-8
Mots-clés : low-lying zero, L-function 
Knightly, Andrew; Reno, Caroline. Weighted Distribution of Low-lying Zeros of GL(2) $L$-functions. Canadian journal of mathematics, Tome 71 (2019) no. 1, pp. 153-182. doi: 10.4153/CJM-2018-013-8
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     title = {Weighted {Distribution} of {Low-lying} {Zeros} of {GL(2)} $L$-functions},
     journal = {Canadian journal of mathematics},
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