Fungrim entry: 3f96c1

Symbol: DirichletLZero —

\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

n

is a nonzero integer such that

n > 0

gives zeros with

\operatorname{Im}\!\left(\rho_{n,\chi}\right) > 0

, ordered by increasing imaginary part, while

n < 0

gives zeros with

\operatorname{Im}\!\left(\rho_{n,\chi}\right) \le 0

, ordered by decreasing imaginary part.

Definitions:

Fungrim symbol	Notation	Short description
`DirichletLZero`	$\rho_{n,\chi}$	Nontrivial zero of Dirichlet L-function
`RiemannZetaZero`	$\rho_{n}$	Nontrivial zero of the Riemann zeta function
`Im`	$\operatorname{Im}(z)$	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."))

Topics using this entry

Dirichlet L-functions