Dirichlet series
| $\zeta_K(s)$ = 1 | + 3-s + 4-s + 9-s + 2·11-s + 12-s + 16-s + 2·17-s + 2·23-s + 25-s + 27-s + 2·33-s + 36-s + 2·44-s + 48-s + 49-s + 2·51-s + 2·61-s + 64-s + 2·67-s + 2·68-s + 2·69-s + 2·71-s + 75-s + 2·79-s + 81-s + 2·89-s + 2·92-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda_K(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\C}(s) \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(723\) = \(3 \cdot 241\) |
| Sign: | $1$ |
| Arithmetic: | yes |
| Primitive: | no |
| Self-dual: | yes |
| Selberg data: | \((2,\ 723,\ (\ :0),\ 1)\) |
Particular Values
\[\zeta_K(1/2) \approx -0.6744060411\]
Pole at \(s=1\)
Euler product
\(\zeta_K(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Factorization
\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\) \(L(s,\chi_{723}(722, \cdot))\)
Imaginary part of the first few zeros on the critical line