Dirichlet series
| $\zeta_K(s)$ = 1 | + 2·2-s + 3·4-s + 2·5-s + 2·7-s + 4·8-s + 9-s + 4·10-s + 4·14-s + 5·16-s + 2·18-s + 2·19-s + 6·20-s + 3·25-s + 6·28-s + 31-s + 6·32-s + 4·35-s + 3·36-s + 4·38-s + 8·40-s + 2·41-s + 2·45-s + 2·47-s + 3·49-s + 6·50-s + 8·56-s + 2·59-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda_K(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, \zeta_K(s)\cr
=\mathstrut & \, \Lambda_K(1-s)
\end{aligned}
\]
Invariants
| \( d \) | = | \(2\) |
| \( N \) | = | \(31\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | no |
| self-dual | : | yes |
| Selberg data | = | $(2,\ 31,\ (\ :0),\ 1)$ |
Euler product
\[\begin{aligned}
\zeta_K(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}
\end{aligned}\]
Factorization
\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\) \(L(s,\chi_{31}(30, \cdot))\)
Particular Values
\[\zeta_K(1/2) \approx -3.225385518\]
Pole at \(s=1\)