Dirichlet series
| L(s) = 1 | − 2.77·2-s − 0.218·3-s + 4.10·4-s + 1.47·5-s + 0.608·6-s − 0.164·7-s − 4.13·8-s + 1.20·9-s − 4.09·10-s + 1.88·11-s − 0.898·12-s − 0.488·13-s + 0.458·14-s − 0.322·15-s + 3.35·16-s − 0.0719·17-s − 3.33·18-s + 0.0726·19-s + 6.04·20-s + 0.0361·21-s − 5.24·22-s + 2.07·23-s + 0.904·24-s − 0.365·25-s + 1.35·26-s − 0.734·27-s − 0.676·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+17.6i) \, \Gamma_{\R}(s+9.90i) \, \Gamma_{\R}(s-17.6i) \, \Gamma_{\R}(s-9.90i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(19.6042\) |
| Root analytic conductor: | \(2.10420\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 1,\ (17.65099838188i, 9.9085043219i, -17.65099838188i, -9.9085043219i:\ ),\ 1)\) |
Euler product
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−24.96158638, −21.56805334, −19.28626345, −17.21913047, −9.46249234, −6.76897170, −1.71999828, −0.97352235, 0.97352235, 1.71999828, 6.76897170, 9.46249234, 17.21913047, 19.28626345, 21.56805334, 24.96158638