Fungrim entry: c2750a

L\!\left(1, \chi\right) = -\frac{1}{q} \sum_{k=1}^{q - 1} \chi(k) \psi\!\left(\frac{k}{q}\right)

Assumptions:

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

TeX:

L\!\left(1, \chi\right) = -\frac{1}{q} \sum_{k=1}^{q - 1} \chi(k) \psi\!\left(\frac{k}{q}\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`DirichletL`	$L\!\left(s, \chi\right)$	Dirichlet L-function
`Sum`	$\sum_{n} f(n)$	Sum
`DigammaFunction`	$\psi\!\left(z\right)$	Digamma function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character

Source code for this entry:

Entry(ID("c2750a"),
    Formula(Equal(DirichletL(1, chi), Neg(Mul(Div(1, q), Sum(Mul(chi(k), DigammaFunction(Div(k, q))), For(k, 1, Sub(q, 1))))))),
    Variables(q, chi),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)), NotEqual(chi, DirichletCharacter(q, 1)))))

Topics using this entry

Dirichlet L-functions