Properties

Degree 2
Conductor $ 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-s + 3·4-s + 2·5-s + 2·6-s + 4·8-s + 9-s + 4·10-s + 2·11-s + 3·12-s + 13-s + 2·15-s + 5·16-s + 2·18-s + 6·20-s + 4·22-s + 4·24-s + 3·25-s + 2·26-s + 27-s + 4·30-s + 6·32-s + 2·33-s + 3·36-s + 39-s + 8·40-s + 2·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda_K(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(39\)    =    \(3 \cdot 13\)
\( \varepsilon \)  =  $1$
primitive  :  no
self-dual  :  yes
Selberg data  =  $(2,\ 39,\ (\ :0),\ 1)$

Euler product

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

Factorization

\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\) \(L(s,\chi_{39}(38, \cdot))\)

Particular Values

\[\zeta_K(1/2) \approx -4.054793217\]
Pole at \(s=1\)

Imaginary part of the first few zeros on the critical line

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