Dirichlet series
| L(s) = 1 | + (0.553 + 0.741i)2-s + (−0.00758 + 0.0796i)3-s + (−0.0630 + 0.820i)4-s + (−0.564 − 0.0667i)5-s + (−0.0632 + 0.0384i)6-s + (1.33 + 0.273i)7-s + (0.00997 − 0.200i)8-s + (−0.504 − 0.00120i)9-s + (−0.263 − 0.455i)10-s + (−0.578 − 0.551i)11-s + (−0.0648 − 0.0112i)12-s + (0.0593 + 0.523i)13-s + (0.534 + 1.13i)14-s + (0.00959 − 0.0444i)15-s + (−0.00150 + 0.0438i)16-s + (−0.108 − 0.782i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+21.0i) \, \Gamma_{\R}(s+1.95i) \, \Gamma_{\R}(s-4.74i) \, \Gamma_{\R}(s-18.2i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.22489\) |
| Root analytic conductor: | \(1.22131\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((4,\ 1,\ (21.0013177272i, 1.95152402253i, -4.74207909766i, -18.21076265204i:\ ),\ 1)\) |
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
−24.22162400, −23.04708087, −20.02707661, −17.71380605, −15.15629091, −13.74444211, −11.75536751, −10.69204911, −7.97514359, −4.91033784, 7.97811213, 11.64270161, 14.04564600, 15.98081824, 20.98395063, 22.63590375, 24.06263678, 24.98876898