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
−24.98876898, −24.06263678, −22.63590375, −20.98395063, −15.98081824, −14.04564600, −11.64270161, −7.97811213, 4.91033784, 7.97514359, 10.69204911, 11.75536751, 13.74444211, 15.15629091, 17.71380605, 20.02707661, 23.04708087, 24.22162400