# Fungrim entry: 6c3fff

$L\!\left(1, \chi\right) = \begin{cases} {\tilde \infty}, & \chi = \chi_{q \, . \, 1}\\\lim_{s \to 1} L\!\left(s, \chi\right), & \text{otherwise}\\ \end{cases}$
Assumptions:$q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q}$
TeX:
L\!\left(1, \chi\right) = \begin{cases} {\tilde \infty}, & \chi = \chi_{q \, . \, 1}\\\lim_{s \to 1} L\!\left(s, \chi\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
UnsignedInfinity${\tilde \infty}$ Unsigned infinity
DirichletCharacter$\chi_{q \, . \, \ell}$ Dirichlet character
ComplexLimit$\lim_{z \to a} f(z)$ Limiting value, complex variable
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("6c3fff"),
Formula(Equal(DirichletL(1, chi), Cases(Tuple(UnsignedInfinity, Equal(chi, DirichletCharacter(q, 1))), Tuple(ComplexLimit(DirichletL(s, chi), For(s, 1)), 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