Dirichlet series
| L(s) = 1 | + (0.340 − 0.484i)2-s + (0.0264 − 0.577i)3-s + (−0.790 − 0.329i)4-s + (1.16 + 0.105i)5-s + (−0.270 − 0.209i)6-s + (0.165 − 0.0772i)7-s + (−0.316 + 1.08i)8-s + (−0.536 − 0.0305i)9-s + (0.447 − 0.528i)10-s + (−0.442 − 0.761i)11-s + (−0.211 + 0.448i)12-s + (0.756 + 0.0103i)13-s + (0.0188 − 0.106i)14-s + (0.0917 − 0.671i)15-s + (0.296 + 0.742i)16-s + (−0.200 + 0.739i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-19.0i) \, \Gamma_{\R}(s-1.97i) \, \Gamma_{\R}(s+5.10i) \, \Gamma_{\R}(s+15.9i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.91798\) |
| Root analytic conductor: | \(1.17682\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((4,\ 1,\ (-19.03767623312i, -1.972435011632i, 5.10355437536i, 15.9065568694i:\ ),\ 1)\) |
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
−23.20477758, −22.06186511, −20.90486526, −17.99614909, −13.51432196, −9.56782573, 5.68994723, 8.76421280, 10.91443624, 13.20797304, 14.09560366, 17.62720365, 21.16901722, 22.76927297, 24.08379404