Dirichlet series
| L(s) = 1 | + (−0.0851 + 1.58i)2-s + (−0.314 + 0.584i)3-s + (−0.353 − 0.269i)4-s + (−1.02 + 0.236i)5-s + (−0.897 − 0.547i)6-s + (0.768 + 0.341i)7-s + (0.188 + 1.25i)8-s + (0.365 − 0.368i)9-s + (−0.286 − 1.63i)10-s + (−0.998 + 0.391i)11-s + (0.268 − 0.122i)12-s + (1.01 + 0.00943i)13-s + (−0.605 + 1.18i)14-s + (0.182 − 0.671i)15-s + (−1.25 − 0.384i)16-s + (0.221 + 0.412i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+20.1i) \, \Gamma_{\R}(s+3.32i) \, \Gamma_{\R}(s-5.45i) \, \Gamma_{\R}(s-18.0i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.21040\) |
| Root analytic conductor: | \(1.43245\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((4,\ 1,\ (20.174166098i, 3.32177319608i, -5.45937323438i, -18.03656605964i:\ ),\ 1)\) |
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
−24.6949831, −23.4129015, −20.9221404, −18.6588134, −15.9469816, −12.9580412, −11.6024029, −10.8229102, −7.7083918, −1.1134283, 7.7192476, 11.2436877, 15.1848943, 15.8495149, 20.9535343, 23.3156179, 23.9923408