Dirichlet series
| $\zeta_K(s)$ = 1 | + 2·2-s + 3-s + 3·4-s + 2·6-s + 2·7-s + 4·8-s + 9-s + 3·12-s + 4·14-s + 5·16-s + 2·18-s + 19-s + 2·21-s + 4·24-s + 25-s + 27-s + 6·28-s + 2·29-s + 6·32-s + 3·36-s + 2·38-s + 2·41-s + 4·42-s + 2·43-s + 5·48-s + 3·49-s + 2·50-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda_K(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\R}(s)^{2} \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(57\) = \(3 \cdot 19\) |
| Sign: | $1$ |
| Arithmetic: | yes |
| Primitive: | no |
| Self-dual: | yes |
| Selberg data: | \((2,\ 57,\ (0, 0:\ ),\ 1)\) |
Particular Values
\[\zeta_K(1/2) \approx -2.022607398\]
Pole at \(s=1\)
Euler product
\(\zeta_K(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Factorization
\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\) \(L(s,\chi_{57}(56, \cdot))\)
Imaginary part of the first few zeros on the critical line