Properties

Degree $2$
Conductor $364$
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 + 2·5-s + 2·6-s + 7-s + 8-s + 3·9-s + 2·10-s + 2·11-s + 2·12-s + 13-s + 14-s + 4·15-s + 16-s + 3·18-s + 2·20-s + 2·21-s + 2·22-s + 2·24-s + 3·25-s + 26-s + 4·27-s + 28-s + 2·29-s + 4·30-s + 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda_K(s)=\mathstrut & 364 ^{s/2} \, \Gamma_{\R}(s)^{2} \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(364\)    =    \(2^{2} \cdot 7 \cdot 13\)
Sign: $1$
Arithmetic: yes
Primitive: no
Self-dual: yes
Selberg data: \((2,\ 364,\ (0, 0:\ ),\ 1)\)

Particular Values

\[\zeta_K(1/2) \approx -3.189480940\]
Pole at \(s=1\)

Euler product

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

Factorization

\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\) \(L(s,\chi_{364}(363, \cdot))\)

Imaginary part of the first few zeros on the critical line

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