{
  "id": "2201.00473",
  "version": "v2",
  "published": "2022-01-03T04:56:17.000Z",
  "updated": "2022-03-17T02:20:46.000Z",
  "title": "Determination of $\\textrm{GL}(3)$ cusp forms by central values of quadratic twisted $L$-functions",
  "authors": [
    "Shenghao Hua",
    "Bingrong Huang"
  ],
  "comment": "23 pages. Incorporate the referees' comments and corrections, accepted in IMRN",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\phi$ and $\\phi'$ be two $\\textrm{GL}(3)$ Hecke--Maass cusp forms. In this paper, we prove that $\\phi=\\phi'\\textrm{ or }\\widetilde{\\phi'}$ if there exists a nonzero constant $\\kappa$ such that $$L(\\frac{1}{2},\\phi\\otimes \\chi_{8d})=\\kappa L(\\frac{1}{2},\\phi'\\otimes \\chi_{8d})$$ for all positive odd square-free positive $d$. Here $\\widetilde{\\phi'}$ is dual form of $\\phi'$ and $\\chi_{8d}$ is the quadratic character $(\\frac{8d}{\\cdot})$. To prove this, we obtain asymptotic formulas for twisted first moment of central values of quadratic twisted $L$-functions on $\\textrm{GL}(3)$, which will have many other applications.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-03-17T02:20:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "central values",
      "determination",
      "hecke-maass cusp forms",
      "positive odd square-free",
      "dual form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}