Properties

Degree $2$
Conductor $308$
Sign $1$
Motivic weight $0$
Primitive no
Self-dual yes

Related objects

Learn more about

Normalization:  

(not yet available)

Dirichlet series

$\zeta_K(s)$  = 1  + 2-s + 2·3-s + 4-s + 2·6-s + 7-s + 8-s + 3·9-s + 11-s + 2·12-s + 2·13-s + 14-s + 16-s + 2·17-s + 3·18-s + 2·21-s + 22-s + 2·24-s + 25-s + 2·26-s + 4·27-s + 28-s + 2·31-s + 32-s + 2·33-s + 2·34-s + 3·36-s + 2·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda_K(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $1$
Primitive: no
Self-dual: yes
Selberg data: \((2,\ 308,\ (\ :0),\ 1)\)

Particular Values

\[\zeta_K(1/2) \approx -3.731641683\]
Pole at \(s=1\)

Euler product

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

Factorization

\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\) \(L(s,\chi_{308}(307, \cdot))\)

Imaginary part of the first few zeros on the critical line

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