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
−22.8650342, −20.3642624, −18.9905560, −17.8583767, −16.9152521, −15.9554273, −13.0949839, −11.4224256, −9.9565801, −8.7739335, −7.6539659, −4.9978665, −0.3963276, 10.6322979, 18.1098191, 19.3345203, 22.3645305, 23.5455547