Dirichlet series
| L(s) = 1 | + 1.45·2-s − 0.139·3-s + 0.750·4-s + 0.963·5-s − 0.203·6-s − 1.78·7-s + 0.574·8-s + 0.476·9-s + 1.39·10-s − 0.899·11-s − 0.105·12-s + 1.31·13-s − 2.58·14-s − 0.134·15-s + 0.921·16-s + 0.0239·17-s + 0.691·18-s − 0.251·19-s + 0.723·20-s + 0.249·21-s − 1.30·22-s − 0.281·23-s − 0.0802·24-s − 0.314·25-s + 1.89·26-s − 0.270·27-s − 1.33·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+18.8i) \, \Gamma_{\R}(s+3.62i) \, \Gamma_{\R}(s-18.8i) \, \Gamma_{\R}(s-3.62i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.97030\) |
| Root analytic conductor: | \(1.31280\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 1,\ (18.84729338548i, 3.62223069428i, -18.84729338548i, -3.62223069428i:\ ),\ 1)\) |
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
−23.450898072, −22.585114992, −21.365376011, −15.972077653, −13.512197480, −13.008093170, −10.069332907, −6.086187077, 6.086187077, 10.069332907, 13.008093170, 13.512197480, 15.972077653, 21.365376011, 22.585114992, 23.450898072