# Fungrim entry: 1bd945

$L\!\left(s, \chi\right) = L\!\left(s, {\chi}_{0}\right) \prod_{p \mid q} \left(1 - \frac{{\chi}_{0}\!\left(p\right)}{{p}^{s}}\right)\; \text{ where } {\chi}_{1} = \chi_{q \, . \, 1},\;\chi = {\chi}_{0} {\chi}_{1}$
Assumptions:$q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; d \in \{1, 2, \ldots, q\} \;\mathbin{\operatorname{and}}\; d \mid q \;\mathbin{\operatorname{and}}\; {\chi}_{0} \in G^{\text{Primitive}}_{d} \;\mathbin{\operatorname{and}}\; s \in \mathbb{C}$
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^{\text{Primitive}}_{d} \;\mathbin{\operatorname{and}}\; s \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
DirichletL$L\!\left(s, \chi\right)$ Dirichlet L-function
PrimeProduct$\prod_{p} f(p)$ Product over primes
Pow${a}^{b}$ Power
DirichletCharacter$\chi_{q \, . \, \ell}$ Dirichlet character
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Range$\{a, a + 1, \ldots, b\}$ Integers between given endpoints
PrimitiveDirichletCharacters$G^{\text{Primitive}}_{q}$ Primitive Dirichlet characters with given modulus
CC$\mathbb{C}$ 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."))

## Topics using this entry

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