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Fungrim entry: ea8c55

Poles ⁣(L ⁣(s,χ),s,C{~})={{1},χ=χq(1,){},otherwise\operatorname{Poles}\!\left(L\!\left(s, \chi\right), s, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \begin{cases} \left\{1\right\}, & \chi = \chi_{q}(1, \cdot)\\\left\{\right\}, & \text{otherwise}\\ \end{cases}
Assumptions:qZ1andχGqq \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q}
\operatorname{Poles}\!\left(L\!\left(s, \chi\right), s, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \begin{cases} \left\{1\right\}, & \chi = \chi_{q}(1, \cdot)\\\left\{\right\}, & \text{otherwise}\\ \end{cases}

q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q}
Fungrim symbol Notation Short description
DirichletLL ⁣(s,χ)L\!\left(s, \chi\right) Dirichlet L-function
CCC\mathbb{C} Complex numbers
UnsignedInfinity~{\tilde \infty} Unsigned infinity
DirichletCharacterχq(,)\chi_{q}(\ell, \cdot) Dirichlet character
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
DirichletGroupGqG_{q} Dirichlet characters with given modulus
Source code for this entry:
    Formula(Equal(Poles(DirichletL(s, chi), s, Union(CC, Set(UnsignedInfinity))), Cases(Tuple(Set(1), Equal(chi, DirichletCharacter(q, 1))), Tuple(Set(), Otherwise)))),
    Variables(q, chi),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-08-19 14:38:23.809000 UTC