The smallest prime in a conjugacy class and the first sign change for automorphic 𝐿-functions

Cho, Peter; Kim, Henry

doi:10.1090/proc/15233

The smallest prime in a conjugacy class and the first sign change for automorphic $L$-functions
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by Peter J. Cho and Henry H. Kim

Proc. Amer. Math. Soc. 149 (2021), 923-933

DOI: https://doi.org/10.1090/proc/15233

Published electronically: December 31, 2020

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Abstract:

Let $K$ be an $S_n$-field. For a nonidentity conjugacy class $C$, define $N_{K,C}$ to be the smallest prime $p$ such that Frob$_p\in C$. By using the observation that $N_{K,C}$ is interpreted as the first prime sign change of the Dirichlet coefficients of automorphic $L$-functions, we improve the known bound on $N_{K,C}$ for $n=3,4,5$. (For $n=5$, we need to assume the strong Artin conjecture.)

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