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Fungrim entry: c2750a

L ⁣(1,χ)=1qk=1q1χ(k)ψ ⁣(kq)L\!\left(1, \chi\right) = -\frac{1}{q} \sum_{k=1}^{q - 1} \chi(k) \psi\!\left(\frac{k}{q}\right)
Assumptions:qZ1andχGqandχχq.1q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q} \,\mathbin{\operatorname{and}}\, \chi \ne \chi_{q \, . \, 1}
L\!\left(1, \chi\right) = -\frac{1}{q} \sum_{k=1}^{q - 1} \chi(k) \psi\!\left(\frac{k}{q}\right)

q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q} \,\mathbin{\operatorname{and}}\, \chi \ne \chi_{q \, . \, 1}
Fungrim symbol Notation Short description
DirichletLL ⁣(s,χ)L\!\left(s, \chi\right) Dirichlet L-function
Sumnf(n)\sum_{n} f(n) Sum
DigammaFunctionψ ⁣(z)\psi\!\left(z\right) Digamma function
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
DirichletGroupGqG_{q} Dirichlet characters with given modulus
DirichletCharacterχq.\chi_{q \, . \, \ell} Dirichlet character
Source code for this entry:
    Formula(Equal(DirichletL(1, chi), Neg(Mul(Div(1, q), Sum(Mul(chi(k), DigammaFunction(Div(k, q))), For(k, 1, Sub(q, 1))))))),
    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.

2019-11-11 15:50:15.016492 UTC