Fungrim home page

Fungrim entry: 288207

Λ ⁣(s,χ)=εΛ ⁣(1s,χ)   where a=1χ(1)2,  ε=Gq ⁣(χ)iaq\Lambda\!\left(s, \chi\right) = \varepsilon \Lambda\!\left(1 - s, \overline{\chi}\right)\; \text{ where } a = \frac{1 - \chi(-1)}{2},\;\varepsilon = \frac{G_{q}\!\left(\chi\right)}{{i}^{a} \sqrt{q}}
Assumptions:qZ1  and  χGqPrimitive  and  sCq \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G^{\text{Primitive}}_{q} \;\mathbin{\operatorname{and}}\; s \in \mathbb{C}
\Lambda\!\left(s, \chi\right) = \varepsilon \Lambda\!\left(1 - s, \overline{\chi}\right)\; \text{ where } a = \frac{1 - \chi(-1)}{2},\;\varepsilon = \frac{G_{q}\!\left(\chi\right)}{{i}^{a} \sqrt{q}}

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G^{\text{Primitive}}_{q} \;\mathbin{\operatorname{and}}\; s \in \mathbb{C}
Fungrim symbol Notation Short description
DirichletLambdaΛ ⁣(s,χ)\Lambda\!\left(s, \chi\right) Completed Dirichlet L-function
Conjugatez\overline{z} Complex conjugate
GaussSumGq ⁣(χ)G_{q}\!\left(\chi\right) Gauss sum
Powab{a}^{b} Power
ConstIii Imaginary unit
Sqrtz\sqrt{z} Principal square root
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
PrimitiveDirichletCharactersGqPrimitiveG^{\text{Primitive}}_{q} Primitive Dirichlet characters with given modulus
CCC\mathbb{C} Complex numbers
Source code for this entry:
    Formula(Equal(DirichletLambda(s, chi), Where(Mul(epsilon, DirichletLambda(Sub(1, s), Conjugate(chi))), Equal(a, Div(Sub(1, chi(-1)), 2)), Equal(epsilon, Div(GaussSum(q, chi), Mul(Pow(ConstI, a), Sqrt(q))))))),
    Variables(q, chi, s),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, PrimitiveDirichletCharacters(q)), Element(s, CC))))

Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC