# Fungrim entry: ea8c55

$\mathop{\operatorname{poles}\,}\limits_{s \in \mathbb{C} \cup \left\{{\tilde \infty}\right\}} L\!\left(s, \chi\right) = \begin{cases} \left\{1\right\}, & \chi = \chi_{q \, . \, 1}\\\left\{\right\}, & \text{otherwise}\\ \end{cases}$
Assumptions:$q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q}$
TeX:
\mathop{\operatorname{poles}\,}\limits_{s \in \mathbb{C} \cup \left\{{\tilde \infty}\right\}} L\!\left(s, \chi\right) = \begin{cases} \left\{1\right\}, & \chi = \chi_{q \, . \, 1}\\\left\{\right\}, & \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$\chi_{q \, . \, \ell}$ 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("ea8c55"),
Formula(Equal(Poles(DirichletL(s, chi), ForElement(s, Union(CC, Set(UnsignedInfinity)))), Cases(Tuple(Set(1), Equal(chi, DirichletCharacter(q, 1))), Tuple(Set(), 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.

2021-03-15 19:12:00.328586 UTC