Properties

Degree 2
Conductor $ 2^{2} \cdot 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 + 3-s + 4-s + 6-s + 8-s + 9-s + 2·11-s + 12-s + 2·13-s + 16-s + 18-s + 2·22-s + 2·23-s + 24-s + 25-s + 2·26-s + 27-s + 32-s + 2·33-s + 36-s + 2·37-s + 2·39-s + 2·44-s + 2·46-s + 2·47-s + 48-s + 49-s + ⋯

Functional equation

\[\begin{aligned} \Lambda_K(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\R}(s)^{2} \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(12\)    =    \(2^{2} \cdot 3\)
\( \varepsilon \)  =  $1$
primitive  :  no
self-dual  :  yes
Selberg data  =  $(2,\ 12,\ (0, 0:\ ),\ 1)$

Euler product

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

Factorization

\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\) \(L(s,\chi_{12}(11, \cdot))\)

Particular Values

\[\zeta_K(1/2) \approx -0.7280699664\]
Pole at \(s=1\)

Imaginary part of the first few zeros on the critical line

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