Dirichlet series
| $\zeta_K(s)$ = 1 | + 4-s + 5-s + 2·7-s + 9-s + 2·13-s + 16-s + 2·17-s + 20-s + 2·23-s + 25-s + 2·28-s + 2·29-s + 2·35-s + 36-s + 2·37-s + 2·41-s + 2·43-s + 45-s + 3·49-s + 2·52-s + 2·59-s + 2·61-s + 2·63-s + 64-s + 2·65-s + 2·68-s + 2·73-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda_K(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s)^{2} \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(1205\) = \(5 \cdot 241\) |
| Sign: | $1$ |
| Primitive: | no |
| Self-dual: | yes |
| Selberg data: | \((2,\ 1205,\ (0, 0:\ ),\ 1)\) |
Particular Values
\[\zeta_K(1/2) \approx -1.182158125\]
Pole at \(s=1\)
Euler product
\(\zeta_K(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Factorization
\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\) \(L(s,\chi_{1205}(1204, \cdot))\)
Imaginary part of the first few zeros on the critical line