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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}\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: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}\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
ZeroszerosP(x)f ⁣(x)\mathop{\operatorname{zeros}\,}\limits_{P\left(x\right)} f\!\left(x\right) Zeros (roots) of function
DirichletLL ⁣(s,χ)L\!\left(s, \chi\right) Dirichlet L-function
CCC\mathbb{C} Complex numbers
ReRe ⁣(z)\operatorname{Re}\!\left(z\right) Real part
SetBuilder{f ⁣(x):P ⁣(x)}\left\{ f\!\left(x\right) : P\!\left(x\right) \right\} Set comprehension
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), 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)))))

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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