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