Fungrim entry: 9ba78a

$\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C},\,\operatorname{Re}\left(s\right) \le 0} L\!\left(s, \chi\right) = \begin{cases} \left\{ -2 n : n \in \mathbb{Z}_{\ge 1} \right\}, & q = 1\\\left\{ -2 n : n \in \mathbb{Z}_{\ge 0} \right\}, & \chi\!\left(-1\right) = 1 \,\mathbin{\operatorname{and}}\, q \ne 1\\\left\{ -2 n - 1 : n \in \mathbb{Z}_{\ge 0} \right\}, & \chi\!\left(-1\right) = -1\\ \end{cases}$
Assumptions:$q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q}^{\text{primitive}}$
TeX:
\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C},\,\operatorname{Re}\left(s\right) \le 0} L\!\left(s, \chi\right) = \begin{cases} \left\{ -2 n : n \in \mathbb{Z}_{\ge 1} \right\}, & q = 1\\\left\{ -2 n : n \in \mathbb{Z}_{\ge 0} \right\}, & \chi\!\left(-1\right) = 1 \,\mathbin{\operatorname{and}}\, q \ne 1\\\left\{ -2 n - 1 : n \in \mathbb{Z}_{\ge 0} \right\}, & \chi\!\left(-1\right) = -1\\ \end{cases}

q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q}^{\text{primitive}}
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
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
PrimitiveDirichletCharacters$G_{q}^{\text{primitive}}$ Primitive Dirichlet characters with given modulus
Source code for this entry:
Entry(ID("9ba78a"),
Formula(Equal(Zeros(DirichletL(s, chi), s, And(Element(s, CC), LessEqual(Re(s), 0))), Cases(Tuple(SetBuilder(Neg(Mul(2, n)), n, Element(n, ZZGreaterEqual(1))), Equal(q, 1)), Tuple(SetBuilder(Neg(Mul(2, n)), n, Element(n, ZZGreaterEqual(0))), And(Equal(chi(-1), 1), Unequal(q, 1))), Tuple(SetBuilder(Sub(Neg(Mul(2, n)), 1), n, Element(n, ZZGreaterEqual(0))), Equal(chi(-1), -1))))),
Variables(q, chi),
Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, PrimitiveDirichletCharacters(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