Dirichlet series
| $\zeta_K(s)$ = 1 | + 2·3-s + 5-s + 8-s + 3·9-s + 3·11-s + 13-s + 2·15-s + 19-s + 2·24-s + 2·25-s + 4·27-s + 29-s + 6·33-s + 37-s + 2·39-s + 40-s + 3·45-s + 3·53-s + 3·55-s + 2·57-s + 64-s + 65-s + 67-s + 71-s + 3·72-s + 4·75-s + 79-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda_K(s)=\mathstrut & 327 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\C}(s) \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(327\) = \(3 \cdot 109\) |
| Sign: | $1$ |
| Arithmetic: | yes |
| Primitive: | no |
| Self-dual: | yes |
| Selberg data: | \((3,\ 327,\ (0:0),\ 1)\) |
Particular Values
\[\zeta_K(1/2) \approx -0.9015133868\]
Pole at \(s=1\)
Euler product
\(\zeta_K(s) = \displaystyle\prod_p \ \prod_{j=1}^{3} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Factorization
\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\)\(L(s, \rho_{2.327.3t2.a.a})\)
Imaginary part of the first few zeros on the critical line