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Fungrim entry: 789ca4

L ⁣(0,χ)=1qk=1qkχ ⁣(k)L\!\left(0, \chi\right) = -\frac{1}{q} \sum_{k=1}^{q} k \chi\!\left(k\right)
Assumptions:qZ1andχGqandχχq(1,)q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q} \,\mathbin{\operatorname{and}}\, \chi \ne \chi_{q}(1, \cdot)
TeX:
L\!\left(0, \chi\right) = -\frac{1}{q} \sum_{k=1}^{q} k \chi\!\left(k\right)

q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q} \,\mathbin{\operatorname{and}}\, \chi \ne \chi_{q}(1, \cdot)
Definitions:
Fungrim symbol Notation Short description
DirichletLL ⁣(s,χ)L\!\left(s, \chi\right) Dirichlet L-function
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
DirichletGroupGqG_{q} Dirichlet characters with given modulus
DirichletCharacterχq(,)\chi_{q}(\ell, \cdot) Dirichlet character
Source code for this entry:
Entry(ID("789ca4"),
    Formula(Equal(DirichletL(0, chi), Mul(Neg(Div(1, q)), Sum(Mul(k, chi(k)), Tuple(k, 1, q))))),
    Variables(q),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)), Unequal(chi, DirichletCharacter(q, 1)))))

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

2019-08-17 11:32:46.829430 UTC