# Fungrim entry: fe4692

$\operatorname{HolomorphicDomain}\!\left(L\!\left(s, \chi\right), s, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \begin{cases} \mathbb{C} \setminus \left\{1\right\}, & \chi = \chi_{q}(1, \cdot)\\\mathbb{C}, & \text{otherwise}\\ \end{cases}$
Assumptions:$q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q}$
TeX:
\operatorname{HolomorphicDomain}\!\left(L\!\left(s, \chi\right), s, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \begin{cases} \mathbb{C} \setminus \left\{1\right\}, & \chi = \chi_{q}(1, \cdot)\\\mathbb{C}, & \text{otherwise}\\ \end{cases}

q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q}
Definitions:
Fungrim symbol Notation Short description
DirichletL$L\!\left(s, \chi\right)$ Dirichlet L-function
CC$\mathbb{C}$ Complex numbers
UnsignedInfinity${\tilde \infty}$ Unsigned infinity
DirichletCharacter$\chi_{q}(\ell, \cdot)$ Dirichlet character
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
DirichletGroup$G_{q}$ Dirichlet characters with given modulus
Source code for this entry:
Entry(ID("fe4692"),
Formula(Equal(HolomorphicDomain(DirichletL(s, chi), s, Union(CC, Set(UnsignedInfinity))), Cases(Tuple(SetMinus(CC, Set(1)), Equal(chi, DirichletCharacter(q, 1))), Tuple(CC, Otherwise)))),
Variables(q, chi),
Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)))))

## Topics using this entry

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

2019-08-21 11:44:15.926409 UTC