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Fungrim entry: 0f96c3

1L ⁣(s,χ)=p(1χ ⁣(p)ps)\frac{1}{L\!\left(s, \chi\right)} = \prod_{p} \left(1 - \frac{\chi\!\left(p\right)}{{p}^{s}}\right)
Assumptions:qZ1andχGqandsCandRe ⁣(s)>1q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q} \,\mathbin{\operatorname{and}}\, s \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(s\right) > 1
\frac{1}{L\!\left(s, \chi\right)} = \prod_{p} \left(1 - \frac{\chi\!\left(p\right)}{{p}^{s}}\right)

q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q} \,\mathbin{\operatorname{and}}\, s \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(s\right) > 1
Fungrim symbol Notation Short description
DirichletLL ⁣(s,χ)L\!\left(s, \chi\right) Dirichlet L-function
PrimeProductpf ⁣(p)\prod_{p} f\!\left(p\right) Product over primes
Powab{a}^{b} Power
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
DirichletGroupGqG_{q} Dirichlet characters with given modulus
CCC\mathbb{C} Complex numbers
ReRe ⁣(z)\operatorname{Re}\!\left(z\right) Real part
Source code for this entry:
    Formula(Equal(Div(1, DirichletL(s, chi)), PrimeProduct(Parentheses(Sub(1, Div(chi(p), Pow(p, s)))), p))),
    Variables(q, chi, s),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)), Element(s, CC), Greater(Re(s), 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