Dirichlet series
| $\zeta_K(s)$ = 1 | + 3-s + 4-s + 2·5-s + 9-s + 2·11-s + 12-s + 2·13-s + 2·15-s + 16-s + 2·17-s + 2·20-s + 23-s + 3·25-s + 27-s + 2·31-s + 2·33-s + 36-s + 2·39-s + 2·44-s + 2·45-s + 48-s + 49-s + 2·51-s + 2·52-s + 2·53-s + 4·55-s + 2·60-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda_K(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s)^{2} \, \zeta_K(s)\cr
=\mathstrut & \, \Lambda_K(1-s)
\end{aligned}
\]
Invariants
| \( d \) | = | \(2\) |
| \( N \) | = | \(69\) = \(3 \cdot 23\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | no |
| self-dual | : | yes |
| Selberg data | = | $(2,\ 69,\ (0, 0:\ ),\ 1)$ |
Euler product
\[\begin{aligned}
\zeta_K(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}
\end{aligned}\]
Factorization
\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\) \(L(s,\chi_{69}(68, \cdot))\)
Particular Values
\[\zeta_K(1/2) \approx -1.020530437\]
Pole at \(s=1\)