Functional Equations Characterize Dirichlet Characters

Ghaith Hiary; Ali Saraeb

Functional Equations Characterize Dirichlet Characters

Number Theory 2026-07-01 v1

Authors: Ghaith Hiary , Ali Saraeb

Abstract

We prove a converse theorem for functional equations of Dirichlet $L$ -functions. Under mild assumptions, we prove that these functional equations for $L$ -series of the form $\sum_{n\ge 1} f(n) n^{-s}$ force the coefficient function $f$ to be a primitive Dirichlet character. Consequently, these functional equations force the existence of an Euler product.

Cite

@article{arxiv.2607.00332,
  title  = {Functional Equations Characterize Dirichlet Characters},
  author = {Ghaith Hiary and Ali Saraeb},
  journal= {arXiv preprint arXiv:2607.00332},
  year   = {2026}
}