Dirichlet series
| $\zeta_K(s)$ = 1 | + 2-s + 4-s + 2·5-s + 8-s + 9-s + 2·10-s + 2·13-s + 16-s + 2·17-s + 18-s + 2·20-s + 3·25-s + 2·26-s + 2·29-s + 32-s + 2·34-s + 36-s + 2·37-s + 2·40-s + 2·41-s + 2·45-s + 49-s + 3·50-s + 2·52-s + 2·53-s + 2·58-s + 2·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda_K(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(4\) = \(2^{2}\) |
| Sign: | $1$ |
| Primitive: | no |
| Self-dual: | yes |
| Selberg data: | \((2,\ 4,\ (\ :0),\ 1)\) |
Particular Values
\[\zeta_K(1/2) \approx -0.9750662300\]
Pole at \(s=1\)
Euler product
\(\zeta_K(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Factorization
\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\) \(L(s,\chi_{4}(3, \cdot))\)
Imaginary part of the first few zeros on the critical line