Statistics for low-lying zeros of symmetric power L-functions in the level aspect
Published in Forum Mathematicum 23, 969—1028 (2011).
Submitted on 26 March 2007.
In this paper, we compute the one-level density and the two-level density for low-lying zeros of some families of symmetric power L-functions in the level aspect. These families are built according to the value of the sign of the functional equation. As a consequence, we completely determine the symmetry types of these families.
The main technical ingredients are some large sieve inequalities for Kloosterman sums and Riemann-von Mangoldt’s explicit formula.
We also compute the moments of the one-level density and produces a new instance for Hughes-Rudnick’s mock Gaussian behaviour. This result relies on heavy combinatorial arguments.
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