Forbidden conductors and sequences of $\pm 1$s

TL;DR

This paper investigates 'forbidden' conductors in the context of L-functions, demonstrating their density in (0,4) and identifying accumulation points, thus advancing understanding of algebraic criteria in number theory.

Contribution

It proves the density of forbidden conductors in (0,4) and identifies their accumulation points, addressing a previously open problem in the study of L-functions.

Findings

Forbidden conductors are dense in (0,4).

Positive accumulation points of rational forbidden conductors are identified.

The study advances understanding of algebraic criteria for L-functions.

Abstract

We study "forbidden" conductors, i.e. numbers q > 0 satisfying algebraic criteria introduced by J. Kaczorowski, A. Perelli and M. Radziejewski [Acta Arith. 210 (2023), 1-21], that cannot be conductors of L-functions of degree 2 from the extended Selberg class. We show that the set of forbidden q is dense in the interval (0,4), solving a problem posed in [Acta Arith. 210 (2023), 1-21]. We also find positive points of accumulation of rational forbidden q.