# Fungrim entry: bc755b

$\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C}} L\!\left(s, \chi\right) = \left(\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C},\,\operatorname{Re}(s) \le 0} L\!\left(s, \chi\right)\right) \cup \left\{ \rho_{n,\chi} : n \in \mathbb{Z} \setminus \left\{0\right\} \right\}$
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}(s) \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$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$ Zeros (roots) of function
DirichletL$L\!\left(s, \chi\right)$ Dirichlet L-function
CC$\mathbb{C}$ Complex numbers
Re$\operatorname{Re}(z)$ Real part
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), ForElement(s, CC)), Union(Parentheses(Zeros(DirichletL(s, chi), For(s), And(Element(s, CC), LessEqual(Re(s), 0)))), Set(DirichletLZero(n, chi), For(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.

2021-03-15 19:12:00.328586 UTC