Dirichlet series
| L(s) = 1 | + (−0.191 + 0.389i)2-s + (0.0834 − 1.28i)3-s + (−0.375 − 0.149i)4-s + (0.557 − 0.0152i)5-s + (0.486 + 0.279i)6-s + (−0.307 + 0.580i)7-s + (−0.0116 − 0.608i)8-s + (−0.0815 − 0.215i)9-s + (−0.100 + 0.220i)10-s + (−0.514 + 0.676i)11-s + (−0.223 + 0.471i)12-s + (0.718 − 0.942i)13-s + (−0.167 − 0.230i)14-s + (0.0268 − 0.719i)15-s + (−0.474 + 0.150i)16-s + (0.690 + 0.132i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-21.2i) \, \Gamma_{\R}(s-1.49i) \, \Gamma_{\R}(s+4.26i) \, \Gamma_{\R}(s+18.4i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.51070\) |
| Root analytic conductor: | \(1.10865\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((4,\ 1,\ (-21.2021329724i, -1.490163268882i, 4.2609146966i, 18.43138154474i:\ ),\ 1)\) |
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
−23.60388982, −21.83756915, −20.85816701, −16.16324808, −13.75368079, −10.77704704, −9.30431542, 6.12063354, 7.76611026, 9.75443634, 12.52574938, 13.45103740, 15.55370998, 17.86686196, 18.69489724, 23.28270845, 24.94483471