Properties

Degree $2$
Conductor $483$
Sign $1$
Motivic weight $0$
Arithmetic yes
Primitive no
Self-dual yes

Related objects

Learn more about

Normalization:  

(not yet available)

Dirichlet series

$\zeta_K(s)$  = 1  + 3-s + 4-s + 7-s + 9-s + 2·11-s + 12-s + 16-s + 2·19-s + 21-s + 23-s + 25-s + 27-s + 28-s + 2·33-s + 36-s + 2·41-s + 2·44-s + 2·47-s + 48-s + 49-s + 2·53-s + 2·57-s + 2·59-s + 2·61-s + 63-s + 64-s + 69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda_K(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $1$
Arithmetic: yes
Primitive: no
Self-dual: yes
Selberg data: \((2,\ 483,\ (\ :0),\ 1)\)

Particular Values

\[\zeta_K(1/2) \approx -1.091829447\]
Pole at \(s=1\)

Euler product

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

Factorization

\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\) \(L(s,\chi_{483}(482, \cdot))\)

Imaginary part of the first few zeros on the critical line

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