Dirichlet series
| L(s) = 1 | − 0.617·2-s + 1.04·3-s + 0.752·4-s − 0.246·5-s − 0.644·6-s + 0.658·7-s − 1.31·8-s + 0.611·9-s + 0.152·10-s + 1.08·11-s + 0.785·12-s + 0.888·13-s − 0.406·14-s − 0.257·15-s + 0.470·16-s − 0.237·17-s − 0.377·18-s + 2.23·19-s − 0.185·20-s + 0.687·21-s − 0.669·22-s + 0.695·23-s − 1.36·24-s − 0.619·25-s − 0.548·26-s + 1.18·27-s + 0.495·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+21.2i) \, \Gamma_{\R}(s+7.20i) \, \Gamma_{\R}(s-21.2i) \, \Gamma_{\R}(s-7.20i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(14.9462\) |
| Root analytic conductor: | \(1.96622\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 1,\ (21.2028027952i, 7.20480401756i, -21.2028027952i, -7.20480401756i:\ ),\ 1)\) |
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
−24.33005131, −20.09744418, −18.12840763, −15.78267352, −14.25933516, −11.62167374, −9.01619022, −3.19698754, −1.34027428, 1.34027428, 3.19698754, 9.01619022, 11.62167374, 14.25933516, 15.78267352, 18.12840763, 20.09744418, 24.33005131