Properties

Degree $2$
Conductor $643$
Sign $1$
Motivic weight $0$
Arithmetic yes
Primitive no
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$\zeta_K(s)$  = 1  + 4-s + 2·7-s + 9-s + 16-s + 2·23-s + 25-s + 2·28-s + 2·29-s + 2·31-s + 36-s + 3·49-s + 2·53-s + 2·63-s + 64-s + 81-s + 2·83-s + 2·89-s + 2·92-s + 2·97-s + 100-s + 2·101-s + 2·112-s + 2·116-s + 121-s + 2·124-s + 2·131-s + 2·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda_K(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\C}(s) \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(643\)
Sign: $1$
Arithmetic: yes
Primitive: no
Self-dual: yes
Selberg data: \((2,\ 643,\ (\ :0),\ 1)\)

Particular Values

\[\zeta_K(1/2) \approx -0.4417792707\]
Pole at \(s=1\)

Euler product

\(\zeta_K(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Factorization

\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\) \(L(s,\chi_{643}(642, \cdot))\)

Imaginary part of the first few zeros on the critical line

Graph of the $Z$-function along the critical line