Properties

Degree 2
Conductor $ 2^{3} $
Sign $1$
Motivic weight 0
Primitive no
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$\zeta_K(s)$  = 1  + 2-s + 4-s + 2·7-s + 8-s + 9-s + 2·14-s + 16-s + 2·17-s + 18-s + 2·23-s + 25-s + 2·28-s + 2·31-s + 32-s + 2·34-s + 36-s + 2·41-s + 2·46-s + 2·47-s + 3·49-s + 50-s + 2·56-s + 2·62-s + 2·63-s + 64-s + 2·68-s + 2·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda_K(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\R}(s)^{2} \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $1$
primitive  :  no
self-dual  :  yes
Selberg data  =  \((2,\ 8,\ (0, 0:\ ),\ 1)\)

Euler product

\[\begin{aligned}\zeta_K(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Factorization

\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\) \(L(s,\chi_{8}(5, \cdot))\)

Particular Values

\[\zeta_K(1/2) \approx -0.5457223778\]
Pole at \(s=1\)

Imaginary part of the first few zeros on the critical line

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