Properties

Degree 3
Conductor $ 2^{3} \cdot 13 $
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  + 2·2-s + 3·4-s + 4·8-s + 11-s + 2·13-s + 5·16-s + 19-s + 2·22-s + 23-s + 4·26-s + 27-s + 29-s + 3·31-s + 6·32-s + 2·38-s + 41-s + 3·44-s + 2·46-s + 6·52-s + 53-s + 2·54-s + 2·58-s + 59-s + 61-s + 6·62-s + 7·64-s + 67-s + ⋯

Functional equation

\[\begin{aligned} \Lambda_K(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\C}(s) \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(104\)    =    \(2^{3} \cdot 13\)
\( \varepsilon \)  =  $1$
primitive  :  no
self-dual  :  yes
Selberg data  =  $(3,\ 104,\ (0:0),\ 1)$

Euler product

\[\begin{aligned} \zeta_K(s) = \prod_p \ \prod_{j=1}^{3} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Particular Values

\[\zeta_K(1/2) \approx -0.9396272552\]
Pole at \(s=1\)

Imaginary part of the first few zeros on the critical line

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