Dirichlet series
| $\zeta_K(s)$ = 1 | + 4-s + 9-s + 16-s + 2·17-s + 2·19-s + 2·23-s + 25-s + 2·29-s + 36-s + 2·37-s + 2·47-s + 49-s + 2·59-s + 64-s + 67-s + 2·68-s + 2·71-s + 2·73-s + 2·76-s + 81-s + 2·83-s + 2·89-s + 2·92-s + 100-s + 2·103-s + 2·107-s + 2·116-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda_K(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s) \, \zeta_K(s)\cr
=\mathstrut & \, \Lambda_K(1-s)
\end{aligned}
\]
Invariants
| \( d \) | = | \(2\) |
| \( N \) | = | \(67\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | no |
| self-dual | : | yes |
| Selberg data | = | $(2,\ 67,\ (\ :0),\ 1)$ |
Euler product
\[\begin{aligned}
\zeta_K(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}
\end{aligned}\]
Factorization
\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\) \(L(s,\chi_{67}(66, \cdot))\)
Particular Values
\[\zeta_K(1/2) \approx -0.5446094529\]
Pole at \(s=1\)