On the least prime ideal and Siegel zeros

Abstract

Let $K$ be a number field, $\mathfrak{q}$ be an integral ideal, and $\mathrm{Cl}(\mathfrak{q})$ be the associated ray class group. Suppose $\mathrm{Cl}(\mathfrak{q})$ possesses a real exceptional character $\psi$, possibly principal, with a Siegel zero $\beta$. For $\mathcal{C}\in \mathrm{Cl}(\mathfrak{q})$ satisfying $\psi (\mathcal{C})=1$, we establish an effective $K$-uniform Linnik-type bound with explicit constants for the least norm of a prime ideal $\mathfrak{p}\in \mathcal{C}$. A special case of this result is related to rational primes represented by certain binary quadratic forms.

25 pages

References in corpus (3)

• Explicit results on the distribution of zeros of Hecke $L$-functions
• Explicit estimates for the zeros of Hecke $L$-functions
• On the average distribution of primes represented by binary quadratic forms

Cited by in corpus (1)

• An explicit bound for the least prime ideal in the Chebotarev density theorem