Dirichlet series
| $\zeta_K(s)$ = 1 | + 2-s + 2·3-s + 4-s + 2·5-s + 2·6-s + 7-s + 8-s + 3·9-s + 2·10-s + 2·12-s + 2·13-s + 14-s + 4·15-s + 16-s + 3·18-s + 2·19-s + 2·20-s + 2·21-s + 2·23-s + 2·24-s + 3·25-s + 2·26-s + 4·27-s + 28-s + 4·30-s + 32-s + 2·35-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda_K(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, \zeta_K(s)\cr
=\mathstrut & \, \Lambda_K(1-s)
\end{aligned}
\]
Invariants
| \( d \) | = | \(2\) |
| \( N \) | = | \(56\) = \(2^{3} \cdot 7\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | no |
| self-dual | : | yes |
| Selberg data | = | $(2,\ 56,\ (\ :0),\ 1)$ |
Euler product
\[\begin{aligned}
\zeta_K(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}
\end{aligned}\]
Factorization
\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\) \(L(s,\chi_{56}(13, \cdot))\)
Particular Values
\[\zeta_K(1/2) \approx -3.574278729\]
Pole at \(s=1\)