{
  "id": "2005.11381",
  "version": "v1",
  "published": "2020-05-22T20:32:44.000Z",
  "updated": "2020-05-22T20:32:44.000Z",
  "title": "Beyond the extended Selberg class: $d_F\\le 1$",
  "authors": [
    "Ravi Raghunathan"
  ],
  "comment": "20 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We will introduce two new classes of Dirichlet series which are monoids under multiplication. The first class $\\mathfrak{A}^{\\#}$ contains both the extended Selberg class $\\mathscr{S}^{\\#}$ of Kaczorowski and Perelli as well as many $L$-functions attached to automorphic representations of ${\\rm GL}_n({\\mathbb A}_K)$, where ${\\mathbb A}_K$ denotes the ad\\`eles over the number field $K$ (these representations need not be unitary or generic). This is in contrast to the class $\\mathscr{S}^{\\#}$ which is smaller and is known to contain, very few of these $L$-functions. The larger class is obtained by weakening the requirement for absolute convergence, allowing a finite number of poles, allowing more general gamma factors and by allowing the series to have trivial zeros to the right of $\\mathrm{Re}(s)=1/2$, while retaining the other axioms of the extended Selberg class. We will classify series in $\\mathfrak{A}^{\\#}$ of degree $d$ when $d\\le 1$ (when $d=1$, we will assume absolute convergence in $\\mathrm{Re}(s)>1$). We will further prove a primitivity result for the $L$-functions of cuspidal eigenforms on ${\\rm GL}_2({\\mathbb A}_{\\mathbb Q})$ and a theorem allowing us to compare the zeros of tensor product $L$-functions of ${\\rm GL}_n({\\mathbb A}_K)$ which cannot be deduced from previous classification results. The second class $\\mathfrak{G}^{\\#}\\subset\\mathfrak{A}^{\\#}$, which also contains $\\mathscr{S}^{\\#}$, more closely models the behaviour of $L$-functions of unitary globally generic representations of ${\\rm GL}_n({\\mathbb A}_K)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-22T20:32:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M41",
      "11F66"
    ],
    "keywords": [
      "extended selberg class",
      "unitary globally generic representations",
      "assume absolute convergence",
      "general gamma factors",
      "tensor product"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}