The highest lowest zero of general L-functions

Stephen D. Miller showed that, assuming the generalized Riemann Hypothesis, every entire \(L\)-function of real archimedian type has a zero in the interval \(\frac12+i t\) with \(-t_0 < t < t_0\), where \(t_0\approx 14.13\) corresponds to the first zero of the Riemann zeta function. We give an...

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Bibliographic Details
Published in:arXiv.org 2014-09
Main Authors: Bober, Jonathan, Conrey, J Brian, Farmer, David W, Fujii, Akio, Koutsoliotas, Sally, Lemurell, Stefan, Rubinstein, Michael, Yoshida, Hiroyuki
Format: Article
Language:English
Online Access:Get full text
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Summary:Stephen D. Miller showed that, assuming the generalized Riemann Hypothesis, every entire \(L\)-function of real archimedian type has a zero in the interval \(\frac12+i t\) with \(-t_0 < t < t_0\), where \(t_0\approx 14.13\) corresponds to the first zero of the Riemann zeta function. We give an example of a self-dual degree-4 \(L\)-function whose first positive imaginary zero is at \(t_1\approx 14.496\). In particular, Miller's result does not hold for general \(L\)-functions. We show that all \(L\)-functions satisfying some additional (conjecturally true) conditions have a zero in the interval \((-t_2,t_2)\) with \(t_2\approx 22.661\).
ISSN:2331-8422