Fungrim entry: fe4692

L\!\left(s, \chi\right) \text{ is holomorphic on } s \in \begin{cases} \mathbb{C} \setminus \left\{1\right\}, & \chi = \chi_{q \, . \, 1}\\\mathbb{C}, & \text{otherwise}\\ \end{cases}

Assumptions:

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

TeX:

L\!\left(s, \chi\right) \text{ is holomorphic on } s \in \begin{cases} \mathbb{C} \setminus \left\{1\right\}, & \chi = \chi_{q \, . \, 1}\\\mathbb{C}, & \text{otherwise}\\ \end{cases}

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

Definitions:

Fungrim symbol	Notation	Short description
`IsHolomorphic`	$f(z) \text{ is holomorphic at } z = c$	Holomorphic predicate
`DirichletL`	$L\!\left(s, \chi\right)$	Dirichlet L-function
`CC`	$\mathbb{C}$	Complex numbers
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character
`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("fe4692"),
    Formula(IsHolomorphic(DirichletL(s, chi), ForElement(s, Cases(Tuple(SetMinus(CC, Set(1)), Equal(chi, DirichletCharacter(q, 1))), Tuple(CC, Otherwise))))),
    Variables(q, chi),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)))))

Topics using this entry

Dirichlet L-functions