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Fungrim entry: fe4692

L ⁣(s,χ) is holomorphic on s{C{1},χ=χq.1C,otherwiseL\!\left(s, \chi\right) \text{ is holomorphic on } s \in \begin{cases} \mathbb{C} \setminus \left\{1\right\}, & \chi = \chi_{q \, . \, 1}\\\mathbb{C}, & \text{otherwise}\\ \end{cases}
Assumptions:qZ1  and  χGqq \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q}
L\!\left(s, \chi\right) \text{ is holomorphic on } s \in \begin{cases} \mathbb{C} \setminus \left\{1\right\}, & \chi = \chi_{q \, . \, 1}\\\mathbb{C}, & \text{otherwise}\\ \end{cases}

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q}
Fungrim symbol Notation Short description
IsHolomorphicf(z) is holomorphic at z=cf(z) \text{ is holomorphic at } z = c Holomorphic predicate
DirichletLL ⁣(s,χ)L\!\left(s, \chi\right) Dirichlet L-function
CCC\mathbb{C} Complex numbers
DirichletCharacterχq.\chi_{q \, . \, \ell} 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(IsHolomorphic(DirichletL(s, chi), ForElement(s, 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.

2021-03-15 19:12:00.328586 UTC