Fungrim entry: 288207

\Lambda\!\left(s, \chi\right) = \varepsilon \Lambda\!\left(1 - s, \overline{\chi}\right)\; \text{ where } a = \frac{1 - \chi(-1)}{2},\;\varepsilon = \frac{G_{q}\!\left(\chi\right)}{{i}^{a} \sqrt{q}}

Assumptions:

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G^{\text{Primitive}}_{q} \;\mathbin{\operatorname{and}}\; s \in \mathbb{C}

TeX:

\Lambda\!\left(s, \chi\right) = \varepsilon \Lambda\!\left(1 - s, \overline{\chi}\right)\; \text{ where } a = \frac{1 - \chi(-1)}{2},\;\varepsilon = \frac{G_{q}\!\left(\chi\right)}{{i}^{a} \sqrt{q}}

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G^{\text{Primitive}}_{q} \;\mathbin{\operatorname{and}}\; s \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`DirichletLambda`	$\Lambda\!\left(s, \chi\right)$	Completed Dirichlet L-function
`Conjugate`	$\overline{z}$	Complex conjugate
`GaussSum`	$G_{q}\!\left(\chi\right)$	Gauss sum
`Pow`	${a}^{b}$	Power
`ConstI`	$i$	Imaginary unit
`Sqrt`	$\sqrt{z}$	Principal square root
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`PrimitiveDirichletCharacters`	$G^{\text{Primitive}}_{q}$	Primitive Dirichlet characters with given modulus
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("288207"),
    Formula(Equal(DirichletLambda(s, chi), Where(Mul(epsilon, DirichletLambda(Sub(1, s), Conjugate(chi))), Equal(a, Div(Sub(1, chi(-1)), 2)), Equal(epsilon, Div(GaussSum(q, chi), Mul(Pow(ConstI, a), Sqrt(q))))))),
    Variables(q, chi, s),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, PrimitiveDirichletCharacters(q)), Element(s, CC))))

Topics using this entry

Dirichlet L-functions