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Fungrim entry: 2a34c3

zerossC,0<Re(s)<1L ⁣(s,χ)={ρn,χ:nZ{0}}\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C},\,0 < \operatorname{Re}\left(s\right) < 1} L\!\left(s, \chi\right) = \left\{ \rho_{n, \chi} : n \in \mathbb{Z} \setminus \left\{0\right\} \right\}
Assumptions:qZ1andχGqq \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q}
\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C},\,0 < \operatorname{Re}\left(s\right) < 1} L\!\left(s, \chi\right) = \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}
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
DirichletLZeroρn,χ\rho_{n, \chi} Nontrivial zero of Dirichlet L-function
ZZZ\mathbb{Z} Integers
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
DirichletGroupGqG_{q} Dirichlet characters with given modulus
Source code for this entry:
    Formula(Equal(Zeros(DirichletL(s, chi), s, And(Element(s, CC), Less(0, Re(s), 1))), SetBuilder(DirichletLZero(n, chi), n, Element(n, SetMinus(ZZ, Set(0)))))),
    Variables(q, chi),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-08-17 11:32:46.829430 UTC