{
  "id": "math/0210313",
  "version": "v1",
  "published": "2002-10-21T01:42:50.000Z",
  "updated": "2002-10-21T01:42:50.000Z",
  "title": "The critical order of certain Hecke L-functions of imaginary quadratic fields",
  "authors": [
    "Chunlei Liu",
    "Lanju Xu"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $-D < -4$ denote a fundamental discriminant which is either odd or divisible by 8, so that the canonical Hecke character of $\\Bbb Q(\\sqrt{-D})$ exists. Let $d$ be a fundamental discriminant prime to $D$. Let $2k-1$ be an odd natural integer prime to the class number of $\\Bbb Q(\\sqrt{-D})$. Let $\\chi$ be the twist of the $(2k-1)$th power of a canonical Hecke character of $\\Bbb Q(\\sqrt{-D})$ by the Kronecker's symbol $n\\mapsto(\\frac{d}{n})$. It is proved that the order of the Hecke $L$-function $L(s,\\chi)$ at its central point $s=k$ is determined by its root number when $|d| \\leq c(\\epsilon)D^{{1/24}-\\epsilon}$ or, when $|d| \\leq c(\\epsilon)D^{\\frac1{12} -\\epsilon}$ and $k\\geq 2$, where $\\epsilon > 0$ and $c(\\epsilon)$ is a constant depending only on $\\epsilon$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-10-21T01:42:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R42",
      "11G05"
    ],
    "keywords": [
      "imaginary quadratic fields",
      "hecke l-functions",
      "critical order",
      "canonical hecke character",
      "odd natural integer prime"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math.....10313L"
    }
  }
}