---
tag: 5VBZBXSFDQ
kind: concept
name: Siegel zero
msc: [11M20, 11M26]
sources:
  - url: https://en.wikipedia.org/wiki/Siegel_zero
same_as:
  wikidata: Q1773226
  wikipedia: Siegel zero
---

## Summary

A Siegel zero, or Landau–Siegel zero, is a hypothetical real zero of a [[Dirichlet L-function|9D2Z543EJQ]] of a real character that lies very close to $1$.
The classical zero-free region for Dirichlet L-functions excludes every zero near $s = 1$ except possibly one such zero.
Whether Siegel zeros exist is open; the [[generalized Riemann hypothesis|DNKA4WBKDS]] rules them out.
Their possible existence is the source of ineffective constants in results such as the [[Siegel–Walfisz theorem|XEPW251A97]].

## Explanations

### L1 human

Dirichlet L-functions are cousins of the Riemann zeta function, one for each Dirichlet character.
Their zeros control how primes are spread over residue classes.
One knows that no zero comes very close to the point $s = 1$, with one possible exception: a single real zero for a real character.
That possible exception is called a Siegel zero.
Nobody has found one, and most experts believe they do not exist, but nobody can rule them out either.