Fungrim entry: 629f70

L\!\left(s, \chi_{q \, . \, 1}\right) = \zeta\!\left(s\right) \prod_{p \mid q} \left(1 - \frac{1}{{p}^{s}}\right)

Assumptions:

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; s \in \mathbb{C}

TeX:

L\!\left(s, \chi_{q \, . \, 1}\right) = \zeta\!\left(s\right) \prod_{p \mid q} \left(1 - \frac{1}{{p}^{s}}\right)

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; s \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`DirichletL`	$L\!\left(s, \chi\right)$	Dirichlet L-function
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`PrimeProduct`	$\prod_{p} f(p)$	Product over primes
`Pow`	${a}^{b}$	Power
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("629f70"),
    Formula(Equal(DirichletL(s, DirichletCharacter(q, 1)), Mul(RiemannZeta(s), PrimeProduct(Parentheses(Sub(1, Div(1, Pow(p, s)))), For(p), Divides(p, q))))),
    Variables(q, s),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(s, CC))))

Topics using this entry

Dirichlet L-functions