# Dirichlet $L$-functions of quadratic characters of prime conductor at the central point

@article{Baluyot2021DirichletO,
title={Dirichlet \$L\$-functions of quadratic characters of prime conductor at the central point},
author={Siegfred Alan C. Baluyot and Kyle Pratt},
journal={Journal of the European Mathematical Society},
year={2021}
}
• Published 26 September 2018
• Mathematics
• Journal of the European Mathematical Society
We prove that more than nine percent of the central values $L(\frac{1}{2},\chi_p)$ are non-zero, where $p\equiv 1 \pmod{8}$ ranges over primes and $\chi_p$ is the real primitive Dirichlet character of conductor $p$. Previously, it was not known whether a positive proportion of these central values are non-zero. As a by-product, we obtain the order of magnitude of the second moment of $L(\frac{1}{2},\chi_p)$, and conditionally we obtain the order of magnitude of the third moment. Assuming the… Expand
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Non-vanishing of Dirichlet L-functions in Galois orbits
• Mathematics
• 2015
A well known result of Iwaniec and Sarnak states that for at least one third of the primitive Dirichlet characters to a large modulus q, the associated L-functions do not vanish at the central point.Expand
Moments and distribution of central $$L$$L-values of quadratic twists of elliptic curves
• Mathematics
• 2014
We show that if one can compute a little more than a particular moment for some family of L-functions, then one has upper bounds of the conjectured order of magnitude for all smaller (positive, real)Expand
Simple zeros of the zeta function of a quadratic number field. I
• Mathematics
• 1986
Let K be a fixed quadratic extension of Q and write ζK(s) for the Dedekind zeta-function of K, where s = σ + it. It is wellknown, and easy to prove, that the number NK(T) of zeros of ζK(s) in theExpand
The non-vanishing of central values of automorphic L-functions and Landau-Siegel zeros
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• 2000
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Explicit upper bound for the (analytic) rank of J0(q)
• Mathematics
• 2000
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Zeroes of zeta functions and symmetry
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Long mollifiers of the Riemann Zeta-function
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