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Fungrim entry: 3f96c1

Symbol: DirichletLZero ρn,χ\rho_{n,\chi} Nontrivial zero of Dirichlet L-function
Generalizing RiemannZetaZero, this gives an enumeration of the nontrivial zeros of a given Dirichlet L-function, where eventual repeated zeros are counted separately. The index nn is a nonzero integer such that n>0n > 0 gives zeros with Im ⁣(ρn,χ)>0\operatorname{Im}\!\left(\rho_{n,\chi}\right) > 0, ordered by increasing imaginary part, while n<0n < 0 gives zeros with Im ⁣(ρn,χ)0\operatorname{Im}\!\left(\rho_{n,\chi}\right) \le 0, ordered by decreasing imaginary part.
Definitions:
Fungrim symbol Notation Short description
DirichletLZeroρn,χ\rho_{n,\chi} Nontrivial zero of Dirichlet L-function
RiemannZetaZeroρn\rho_{n} Nontrivial zero of the Riemann zeta function
ImIm ⁣(z)\operatorname{Im}\!\left(z\right) Imaginary part
Source code for this entry:
Entry(ID("3f96c1"),
    SymbolDefinition(DirichletLZero, DirichletLZero(n, chi), "Nontrivial zero of Dirichlet L-function"),
    Description("Generalizing", SourceForm(RiemannZetaZero), ", this gives an enumeration of the nontrivial zeros of a given Dirichlet L-function, where eventual repeated zeros are counted separately.", "The index", n, "is a nonzero integer such that", Greater(n, 0), "gives zeros with", Greater(Im(DirichletLZero(n, chi)), 0), ", ordered by increasing imaginary part, while", Less(n, 0), "gives zeros with", LessEqual(Im(DirichletLZero(n, chi)), 0), ", ordered by decreasing imaginary part."))

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

2019-09-16 21:17:18.797188 UTC