Fungrim home page

Fungrim entry: 9ba78a

zerossC,Re(s)0L ⁣(s,χ)={{2n:nZ1},q=1{2n:nZ0},χ(1)=1andq1{2n1:nZ0},χ(1)=1\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C},\,\operatorname{Re}(s) \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(-1) = 1 \,\mathbin{\operatorname{and}}\, q \ne 1\\\left\{ -2 n - 1 : n \in \mathbb{Z}_{\ge 0} \right\}, & \chi(-1) = -1\\ \end{cases}
Assumptions:qZ1andχGqprimitiveq \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}(s) \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(-1) = 1 \,\mathbin{\operatorname{and}}\, q \ne 1\\\left\{ -2 n - 1 : n \in \mathbb{Z}_{\ge 0} \right\}, & \chi(-1) = -1\\ \end{cases}

q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q}^{\text{primitive}}
Definitions:
Fungrim symbol Notation Short description
ZeroszerosxSf(x)\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x) Zeros (roots) of function
DirichletLL ⁣(s,χ)L\!\left(s, \chi\right) Dirichlet L-function
CCC\mathbb{C} Complex numbers
ReRe(z)\operatorname{Re}(z) Real part
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
PrimitiveDirichletCharactersGqprimitiveG_{q}^{\text{primitive}} Primitive Dirichlet characters with given modulus
Source code for this entry:
Entry(ID("9ba78a"),
    Formula(Equal(Zeros(DirichletL(s, chi), For(s), And(Element(s, CC), LessEqual(Re(s), 0))), Cases(Tuple(Set(Neg(Mul(2, n)), For(n), Element(n, ZZGreaterEqual(1))), Equal(q, 1)), Tuple(Set(Neg(Mul(2, n)), For(n), Element(n, ZZGreaterEqual(0))), And(Equal(chi(-1), 1), Unequal(q, 1))), Tuple(Set(Sub(Neg(Mul(2, n)), 1), For(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.

2019-10-05 13:11:19.856591 UTC