Fungrim entry: 6c3fff

L\!\left(1, \chi\right) = \begin{cases} {\tilde \infty}, & \chi = \chi_{q \, . \, 1}\\\lim_{s \to 1} L\!\left(s, \chi\right), & \text{otherwise}\\ \end{cases}

Assumptions:

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q}

TeX:

L\!\left(1, \chi\right) = \begin{cases} {\tilde \infty}, & \chi = \chi_{q \, . \, 1}\\\lim_{s \to 1} L\!\left(s, \chi\right), & \text{otherwise}\\ \end{cases}

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q}

Definitions:

Fungrim symbol	Notation	Short description
`DirichletL`	$L\!\left(s, \chi\right)$	Dirichlet L-function
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character
`ComplexLimit`	$\lim_{z \to a} f(z)$	Limiting value, complex variable
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus

Source code for this entry:

Entry(ID("6c3fff"),
    Formula(Equal(DirichletL(1, chi), Cases(Tuple(UnsignedInfinity, Equal(chi, DirichletCharacter(q, 1))), Tuple(ComplexLimit(DirichletL(s, chi), For(s, 1)), Otherwise)))),
    Variables(q, chi),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)))))

Topics using this entry

Dirichlet L-functions