Dirichlet series
| $\zeta_K(s)$ = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 9-s + 10-s + 14-s + 16-s + 2·17-s + 18-s + 2·19-s + 20-s + 25-s + 28-s + 32-s + 2·34-s + 35-s + 36-s + 2·37-s + 2·38-s + 40-s + 2·43-s + 45-s + 2·47-s + 49-s + 50-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda_K(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(280\) = \(2^{3} \cdot 5 \cdot 7\) |
| Sign: | $1$ |
| Arithmetic: | yes |
| Primitive: | no |
| Self-dual: | yes |
| Selberg data: | \((2,\ 280,\ (\ :0),\ 1)\) |
Particular Values
\[\zeta_K(1/2) \approx -1.513384642\]
Pole at \(s=1\)
Euler product
\(\zeta_K(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Factorization
\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\) \(L(s,\chi_{280}(69, \cdot))\)
Imaginary part of the first few zeros on the critical line