Dirichlet series
| $\zeta_K(s)$ = 1 | + 5-s + 7-s + 8-s + 11-s + 17-s + 19-s + 2·23-s + 2·25-s + 27-s + 35-s + 37-s + 40-s + 43-s + 2·49-s + 53-s + 55-s + 56-s + 3·59-s + 61-s + 64-s + 67-s + 77-s + 79-s + 83-s + 85-s + 88-s + 89-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda_K(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\C}(s) \, \zeta_K(s)\cr
=\mathstrut & \, \Lambda_K(1-s)
\end{aligned}
\]
Invariants
| \( d \) | = | \(3\) |
| \( N \) | = | \(23\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | no |
| self-dual | : | yes |
| Selberg data | = | $(3,\ 23,\ (0:0),\ 1)$ |
Euler product
\[\begin{aligned}
\zeta_K(s) = \prod_p \ \prod_{j=1}^{3} (1 - \alpha_{j,p}\, p^{-s})^{-1}
\end{aligned}\]
Factorization
\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\)\(L(s, \rho_{2.23.3t2.1c1})\)
Particular Values
\[\zeta_K(1/2) \approx -0.2541547348\]
Pole at \(s=1\)