{
  "id": "2403.13895",
  "version": "v1",
  "published": "2024-03-20T18:06:37.000Z",
  "updated": "2024-03-20T18:06:37.000Z",
  "title": "Quotients of $L$-functions: degrees $n$ and $n-2$",
  "authors": [
    "Ravi Raghunathan"
  ],
  "comment": "38",
  "categories": [
    "math.NT"
  ],
  "abstract": "If $L(s,\\pi)$ and $L(s,\\rho)$ are the Dirichlet series attached to cuspidal automorphic representations $\\pi$ and $\\rho$ of ${\\rm GL}_n({\\mathbb A}_{\\mathbb Q})$ and ${\\rm GL}_{n-2}({\\mathbb A}_{\\mathbb Q})$ respectively, we show that $F_2(s)=L(s,\\pi)/L(s,\\rho)$ has infinitely many poles. We also establish analogous results for Artin $L$-functions and other $L$-functions not yet proven to be automorphic. Using the classification theorems of \\cite{Ragh20} and \\cite{BaRa20}, we show that cuspidal $L$-functions of ${\\rm GL}_3({\\mathbb A}_{\\mathbb Q})$ are primitive in ${\\mathfrak G}$, a monoid that contains both the Selberg class ${\\mathcal{S}}$ and $L(s,\\sigma)$ for all unitary cuspidal automorphic representations $\\sigma$ of ${\\rm GL}_n({\\mathbb A}_{\\mathbb Q})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-20T18:06:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F66",
      "11M41",
      "11F70"
    ],
    "keywords": [
      "unitary cuspidal automorphic representations",
      "dirichlet series",
      "classification theorems",
      "selberg class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}