Dirichlet series
| $\zeta_K(s)$ = 1 | + 2-s + 2·3-s + 4-s + 5-s + 2·6-s + 8-s + 3·9-s + 10-s + 2·12-s + 2·13-s + 2·15-s + 16-s + 3·18-s + 20-s + 2·24-s + 25-s + 2·26-s + 4·27-s + 2·30-s + 2·31-s + 32-s + 3·36-s + 2·37-s + 4·39-s + 40-s + 2·41-s + 2·43-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda_K(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\R}(s)^{2} \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(40\) = \(2^{3} \cdot 5\) |
| Sign: | $1$ |
| Arithmetic: | yes |
| Primitive: | no |
| Self-dual: | yes |
| Selberg data: | \((2,\ 40,\ (0, 0:\ ),\ 1)\) |
Particular Values
\[\zeta_K(1/2) \approx -1.420178477\]
Pole at \(s=1\)
Euler product
\(\zeta_K(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Factorization
\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\) \(L(s,\chi_{40}(29, \cdot))\)
Imaginary part of the first few zeros on the critical line