Properties

Degree $2$
Conductor $68$
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-s + 2·3-s + 4-s + 2·6-s + 2·7-s + 8-s + 3·9-s + 2·11-s + 2·12-s + 2·13-s + 2·14-s + 16-s + 17-s + 3·18-s + 4·21-s + 2·22-s + 2·23-s + 2·24-s + 25-s + 2·26-s + 4·27-s + 2·28-s + 2·31-s + 32-s + 4·33-s + 34-s + 3·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda_K(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(68\)    =    \(2^{2} \cdot 17\)
Sign: $1$
Arithmetic: yes
Primitive: no
Self-dual: yes
Selberg data: \((2,\ 68,\ (\ :0),\ 1)\)

Particular Values

\[\zeta_K(1/2) \approx -3.296240628\]
Pole at \(s=1\)

Euler product

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

Factorization

\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\) \(L(s,\chi_{68}(67, \cdot))\)

Imaginary part of the first few zeros on the critical line

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