Properties

Degree $2$
Conductor $683$
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  + 2·3-s + 4-s + 3·9-s + 2·12-s + 16-s + 2·19-s + 25-s + 4·27-s + 2·29-s + 3·36-s + 2·48-s + 49-s + 2·53-s + 4·57-s + 2·59-s + 2·61-s + 64-s + 2·67-s + 2·71-s + 2·75-s + 2·76-s + 5·81-s + 2·83-s + 4·87-s + 2·97-s + 100-s + 2·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda_K(s)=\mathstrut & 683 ^{s/2} \, \Gamma_{\C}(s) \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(683\)
Sign: $1$
Arithmetic: yes
Primitive: no
Self-dual: yes
Selberg data: \((2,\ 683,\ (\ :0),\ 1)\)

Particular Values

\[\zeta_K(1/2) \approx -0.9944874232\]
Pole at \(s=1\)

Euler product

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

Factorization

\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\) \(L(s,\chi_{683}(682, \cdot))\)

Imaginary part of the first few zeros on the critical line

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