Dirichlet series
| $\zeta_K(s)$ = 1 | + 4-s + 2·5-s + 2·7-s + 9-s + 2·11-s + 16-s + 2·17-s + 19-s + 2·20-s + 2·23-s + 3·25-s + 2·28-s + 4·35-s + 36-s + 2·43-s + 2·44-s + 2·45-s + 2·47-s + 3·49-s + 4·55-s + 2·61-s + 2·63-s + 64-s + 2·68-s + 2·73-s + 76-s + 4·77-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda_K(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, \zeta_K(s)\cr
=\mathstrut & \, \Lambda_K(1-s)
\end{aligned}
\]
Invariants
| \( d \) | = | \(2\) |
| \( N \) | = | \(19\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | no |
| self-dual | : | yes |
| Selberg data | = | $(2,\ 19,\ (\ :0),\ 1)$ |
Euler product
\[\begin{aligned}
\zeta_K(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}
\end{aligned}\]
Factorization
\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\) \(L(s,\chi_{19}(18, \cdot))\)
Particular Values
\[\zeta_K(1/2) \approx -1.174737789\]
Pole at \(s=1\)