{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nThere is an absolute constant $c \u003e 0$ with the following property.\nLet $q \\ge 3$ be an integer and $\\chi$ a [[Dirichlet character|7XB5VJKDPG]] modulo $q$.\nThen the [[Dirichlet L-function|9D2Z543EJQ]] $L(s, \\chi)$ has no zero $s = \\sigma + it$ in the region\n$$\\sigma \\ge 1 - \\frac{c}{\\log\\big(q(\\abs{t} + 2)\\big)}.$$\nThe one possible exception is a single zero when $\\chi$ is a real nonprincipal character; such a zero is real and simple.\nFix such a constant $c$.\nA *Siegel zero* (or *exceptional zero*) for the modulus $q$ is a real number $\\beta$ with\n$$1 - \\frac{c}{\\log q} \u003c \\beta \u003c 1 \\quad\\text{and}\\quad L(\\beta, \\chi) = 0$$\nfor some real nonprincipal character $\\chi$ modulo $q$.\n\n## Remarks\n\nNo Siegel zero is known, so there is no example to check.\nIf the [[generalized Riemann hypothesis|DNKA4WBKDS]] holds, every zero of $L(s, \\chi)$ in the strip $0 \u003c \\operatorname{Re} s \u003c 1$ has real part $1/2$, so for large $q$ no Siegel zero exists.\nA non-example: a zero of $L(s, \\chi)$ for a character $\\chi$ that is not real is never a Siegel zero, because the zero-free region has no exception for such $\\chi$.\nSiegel's theorem bounds a possible exceptional zero: for every $\\varepsilon \u003e 0$ there is $c(\\varepsilon) \u003e 0$ with $\\beta \u003c 1 - c(\\varepsilon) q^{-\\varepsilon}$, but $c(\\varepsilon)$ is not effectively computable.\nThe notion depends on the choice of $c$; statements about Siegel zeros are usually about a sequence of moduli $q$ with $(1 - \\beta)\\log q \\to 0$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "5VBZBXSFDQ",
    "kind": "definition",
    "msc": [
      "11M20",
      "11M26"
    ],
    "name": "Siegel zero",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Siegel_zero"
      }
    ],
    "tag": "SHFJH5JZ5P",
    "uses": [
      "7XB5VJKDPG",
      "9D2Z543EJQ",
      "DNKA4WBKDS"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/SHFJH5JZ5P/",
  "links": [
    {
      "field": "defines",
      "target": "5VBZBXSFDQ"
    },
    {
      "field": "uses",
      "target": "7XB5VJKDPG"
    },
    {
      "field": "uses",
      "target": "9D2Z543EJQ"
    },
    {
      "field": "uses",
      "target": "DNKA4WBKDS"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/SHFJH5JZ5P.md",
  "msc": [
    "11M20",
    "11M26"
  ],
  "name": "Siegel zero",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/SHF/SHFJH5JZ5P.md",
  "status": null,
  "tag": "SHFJH5JZ5P",
  "uri": "m2://SHFJH5JZ5P",
  "url": "https://tamnd.github.io/math2/wiki/siegel-zero-definition/",
  "writing": {
    "tag": "SHFJH5JZ5P",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}