Dirichlet series
| $\zeta_K(s)$ = 1 | + 3-s + 4-s + 2·5-s + 7-s + 9-s + 12-s + 2·15-s + 16-s + 2·17-s + 2·20-s + 21-s + 3·25-s + 27-s + 28-s + 2·35-s + 36-s + 2·37-s + 2·41-s + 2·43-s + 2·45-s + 2·47-s + 48-s + 49-s + 2·51-s + 2·59-s + 2·60-s + 63-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda_K(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\R}(s)^{2} \, \zeta_K(s)\cr
=\mathstrut & \, \Lambda_K(1-s)
\end{aligned}
\]
Invariants
| \( d \) | = | \(2\) |
| \( N \) | = | \(21\) = \(3 \cdot 7\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | no |
| self-dual | : | yes |
| Selberg data | = | $(2,\ 21,\ (0, 0:\ ),\ 1)$ |
Euler product
\[\begin{aligned}
\zeta_K(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}
\end{aligned}\]
Factorization
\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\) \(L(s,\chi_{21}(20, \cdot))\)
Particular Values
\[\zeta_K(1/2) \approx -0.7261793594\]
Pole at \(s=1\)