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Fungrim entry: bc755b

zerossCL ⁣(s,χ)=(zerossC,Re(s)0L ⁣(s,χ)){ρn,χ:nZ{0}}\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C}} L\!\left(s, \chi\right) = \left(\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C},\,\operatorname{Re}\left(s\right) \le 0} L\!\left(s, \chi\right)\right) \cup \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}
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}\left(s\right) \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
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:
Entry(ID("bc755b"),
    Formula(Equal(Zeros(DirichletL(s, chi), s, Element(s, CC)), Union(Parentheses(Zeros(DirichletL(s, chi), s, And(Element(s, CC), LessEqual(Re(s), 0)))), 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