DirichletL

DirichletL[k,j,s]

gives the Dirichlet L-function for the Dirichlet character with modulus k and index j.

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • For , , where is a Dirichlet character.
  • The possible Dirichlet characters modulo k are specified by an index j, and given by DirichletCharacter[k,j,n].
  • DirichletL[k,j,s] can be evaluated to arbitrary numerical precision for integer k and j, and arbitrary complex s.
  • DirichletL automatically threads over lists.

Examples

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Basic Examples  (2)

A plot along the line 1+ ω:

Scope  (4)

Evaluate for complex arguments:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Simple exact values are generated automatically:

DirichletL threads element-wise over lists and matrices:

Applications  (4)

Plot the real part of a DirichletL function on the critical line:

Plot the real part across the critical strip:

Find a zero of a DirichletL function:

Plot real and imaginary parts in the vicinity of nearby zeros:

Neat Examples  (1)

The real part of a Dirichlet L-function:

Introduced in 2008
 (7.0)