Dirichlet series
| $\zeta_K(s)$ = 1 | + 2·2-s + 3-s + 3·4-s + 2·5-s + 2·6-s + 4·8-s + 9-s + 4·10-s + 2·11-s + 3·12-s + 13-s + 2·15-s + 5·16-s + 2·18-s + 6·20-s + 4·22-s + 4·24-s + 3·25-s + 2·26-s + 27-s + 4·30-s + 6·32-s + 2·33-s + 3·36-s + 39-s + 8·40-s + 2·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda_K(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(39\) = \(3 \cdot 13\) |
| Sign: | $1$ |
| Arithmetic: | yes |
| Primitive: | no |
| Self-dual: | yes |
| Selberg data: | \((2,\ 39,\ (\ :0),\ 1)\) |
Particular Values
\[\zeta_K(1/2) \approx -4.054793217\]
Pole at \(s=1\)
Euler product
\(\zeta_K(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Factorization
\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\) \(L(s,\chi_{39}(38, \cdot))\)
Imaginary part of the first few zeros on the critical line