Dirichlet series
| L(s) = 1 | + (−1.91 − 0.0377i)2-s + (−0.799 + 0.0323i)3-s + (1.08 + 0.144i)4-s + (−0.606 − 0.961i)5-s + (1.52 − 0.0317i)6-s + (−0.242 − 0.0323i)7-s + (0.916 − 0.181i)8-s + (−0.134 − 0.0517i)9-s + (1.12 + 1.85i)10-s + (−0.976 + 0.0482i)11-s + (−0.872 − 0.0799i)12-s + (−0.137 − 0.249i)13-s + (0.462 + 0.0709i)14-s + (0.516 + 0.748i)15-s + (−1.88 − 0.0562i)16-s + (−0.292 − 0.852i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-24.6i) \, \Gamma_{\R}(s+0.0621i) \, \Gamma_{\R}(s+7.93i) \, \Gamma_{\R}(s+16.6i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.0646165\) |
| Root analytic conductor: | \(0.504180\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((4,\ 1,\ (-24.625120245i, 0.062191724021i, 7.93038878332i, 16.63253973772i:\ ),\ 1)\) |
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
−23.5455547, −22.3645305, −19.3345203, −18.1098191, −10.6322979, 0.3963276, 4.9978665, 7.6539659, 8.7739335, 9.9565801, 11.4224256, 13.0949839, 15.9554273, 16.9152521, 17.8583767, 18.9905560, 20.3642624, 22.8650342