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

Dirichlet L-functions