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Fungrim entry: 1bd945

L ⁣(s,χ)=L ⁣(s,χ0)pq(1χ0 ⁣(p)ps)   where χ1=χq.1,χ=χ0χ1L\!\left(s, \chi\right) = L\!\left(s, {\chi}_{0}\right) \prod_{p \mid q} \left(1 - \frac{{\chi}_{0}\!\left(p\right)}{{p}^{s}}\right)\; \text{ where } {\chi}_{1} = \chi_{q \, . \, 1},\,\chi = {\chi}_{0} {\chi}_{1}
Assumptions:qZ1andd{1,2,,q}anddqandχ0GdprimitiveandsCq \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, d \in \{1, 2, \ldots, q\} \,\mathbin{\operatorname{and}}\, d \mid q \,\mathbin{\operatorname{and}}\, {\chi}_{0} \in G_{d}^{\text{primitive}} \,\mathbin{\operatorname{and}}\, s \in \mathbb{C}
This allows an L-function of a non-primitive character to be expressed in terms of an L-function of a primitive character.
TeX:
L\!\left(s, \chi\right) = L\!\left(s, {\chi}_{0}\right) \prod_{p \mid q} \left(1 - \frac{{\chi}_{0}\!\left(p\right)}{{p}^{s}}\right)\; \text{ where } {\chi}_{1} = \chi_{q \, . \, 1},\,\chi = {\chi}_{0} {\chi}_{1}

q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, d \in \{1, 2, \ldots, q\} \,\mathbin{\operatorname{and}}\, d \mid q \,\mathbin{\operatorname{and}}\, {\chi}_{0} \in G_{d}^{\text{primitive}} \,\mathbin{\operatorname{and}}\, s \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
DirichletLL ⁣(s,χ)L\!\left(s, \chi\right) Dirichlet L-function
PrimeProductpf(p)\prod_{p} f(p) Product over primes
Powab{a}^{b} Power
DirichletCharacterχq.\chi_{q \, . \, \ell} Dirichlet character
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Range{a,a+1,,b}\{a, a + 1, \ldots, b\} Integers between given endpoints
PrimitiveDirichletCharactersGqprimitiveG_{q}^{\text{primitive}} Primitive Dirichlet characters with given modulus
CCC\mathbb{C} Complex numbers
Source code for this entry:
Entry(ID("1bd945"),
    Formula(Where(Equal(DirichletL(s, chi), Mul(DirichletL(s, Subscript(chi, 0)), PrimeProduct(Parentheses(Sub(1, Div(Call(Subscript(chi, 0), p), Pow(p, s)))), For(p), Divides(p, q)))), Equal(Subscript(chi, 1), DirichletCharacter(q, 1)), Equal(chi, Mul(Subscript(chi, 0), Subscript(chi, 1))))),
    Variables(q, d, Subscript(chi, 0), s),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(d, Range(1, q)), Divides(d, q), Element(Subscript(chi, 0), PrimitiveDirichletCharacters(d)), Element(s, CC))),
    Description("This allows an L-function of a non-primitive character to be expressed in terms of an L-function of a primitive character."))

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

2019-10-05 13:11:19.856591 UTC