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Fungrim entry: fe4692

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

q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q}
Definitions:
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:
Entry(ID("fe4692"),
    Formula(Equal(HolomorphicDomain(DirichletL(s, chi), s, Union(CC, Set(UnsignedInfinity))), Cases(Tuple(SetMinus(CC, Set(1)), Equal(chi, DirichletCharacter(q, 1))), Tuple(CC, 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.

2019-08-21 11:44:15.926409 UTC