# 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:$q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q}^{\text{primitive}} \;\mathbin{\operatorname{and}}\; s \in \mathbb{C}$
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_{q}^{\text{primitive}} \;\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$G_{q}^{\text{primitive}}$ 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

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

2020-08-27 09:56:25.682319 UTC