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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}(s) \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}(s) \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
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
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), ForElement(s, CC)), Union(Parentheses(Zeros(DirichletL(s, chi), For(s), And(Element(s, CC), LessEqual(Re(s), 0)))), Set(DirichletLZero(n, chi), For(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.

2019-12-11 23:01:54.699850 UTC