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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:qZ1andχGqprimitiveandsCq \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q}^{\text{primitive}} \,\mathbin{\operatorname{and}}\, s \in \mathbb{C}
TeX:
\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_{q}^{\text{primitive}} \,\mathbin{\operatorname{and}}\, s \in \mathbb{C}
Definitions:
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_{q}^{\text{primitive}} Primitive Dirichlet characters with given modulus
CCC\mathbb{C} Complex numbers
Source code for this entry:
Entry(ID("288207"),
    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))))

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2019-10-05 13:11:19.856591 UTC