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One-Level Density of Low-lying Zeros of Quadratic and Quartic Hecke
$L$-functions
- Part of
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- Journal:
- Canadian Journal of Mathematics / Volume 72 / Issue 2 / April 2020
- Published online by Cambridge University Press:
- 30 August 2019, pp. 427-454
- Print publication:
- April 2020
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In this paper we prove some one-level density results for the low-lying zeros of families of quadratic and quartic Hecke
$L$-functions of the Gaussian field. As corollaries, we deduce that at least 94.27% and 5%, respectively, of the members of the quadratic family and the quartic family do not vanish at the central point.
Weighted Distribution of Low-lying Zeros of GL(2)
$L$-functions
- Part of
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- Journal:
- Canadian Journal of Mathematics / Volume 71 / Issue 1 / February 2019
- Published online by Cambridge University Press:
- 08 January 2019, pp. 153-182
- Print publication:
- February 2019
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- Article
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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).
