Properties

Degree $2$
Conductor $4353$
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  + 2·2-s + 3-s + 3·4-s + 2·6-s + 4·8-s + 9-s + 3·12-s + 5·16-s + 2·17-s + 2·18-s + 2·23-s + 4·24-s + 25-s + 27-s + 6·32-s + 4·34-s + 3·36-s + 2·43-s + 4·46-s + 5·48-s + 49-s + 2·50-s + 2·51-s + 2·53-s + 2·54-s + 2·59-s + 2·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda_K(s)=\mathstrut & 4353 ^{s/2} \, \Gamma_{\R}(s)^{2} \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4353\)    =    \(3 \cdot 1451\)
Sign: $1$
Arithmetic: yes
Primitive: no
Self-dual: yes
Selberg data: \((2,\ 4353,\ (0, 0:\ ),\ 1)\)

Particular Values

\[\zeta_K(1/2) \approx -2.638282809\]
Pole at \(s=1\)

Euler product

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

Factorization

\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\) \(L(s,\chi_{4353}(4352, \cdot))\)

Imaginary part of the first few zeros on the critical line

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