# Fungrim entry: bc755b

$\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C}} L\!\left(s, \chi\right) = \left(\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C},\,\operatorname{Re}\left(s\right) \le 0} L\!\left(s, \chi\right)\right) \cup \left\{ \rho_{n, \chi} : n \in \mathbb{Z} \setminus \left\{0\right\} \right\}$
Assumptions:$q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q}$
TeX:
\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C}} L\!\left(s, \chi\right) = \left(\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C},\,\operatorname{Re}\left(s\right) \le 0} L\!\left(s, \chi\right)\right) \cup \left\{ \rho_{n, \chi} : n \in \mathbb{Z} \setminus \left\{0\right\} \right\}

q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q}
Definitions:
Fungrim symbol Notation Short description
Zeros$\mathop{\operatorname{zeros}\,}\limits_{P\left(x\right)} f\!\left(x\right)$ Zeros (roots) of function
DirichletL$L\!\left(s, \chi\right)$ Dirichlet L-function
CC$\mathbb{C}$ Complex numbers
Re$\operatorname{Re}\!\left(z\right)$ Real part
SetBuilder$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$ Set comprehension
DirichletLZero$\rho_{n, \chi}$ Nontrivial zero of Dirichlet L-function
ZZ$\mathbb{Z}$ Integers
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("bc755b"),
Formula(Equal(Zeros(DirichletL(s, chi), s, Element(s, CC)), Union(Parentheses(Zeros(DirichletL(s, chi), s, And(Element(s, CC), LessEqual(Re(s), 0)))), SetBuilder(DirichletLZero(n, chi), n, Element(n, SetMinus(ZZ, Set(0))))))),
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.

2019-08-17 11:32:46.829430 UTC