Dirichlet series
| L(s) = 1 | + (0.227 − 2.03i)2-s + (−0.235 + 0.997i)3-s + (−1.89 − 0.926i)4-s + (−0.0482 − 0.303i)5-s + (1.98 + 0.707i)6-s + (−0.207 + 0.290i)7-s + (−1.58 + 1.17i)8-s + (−0.263 − 0.470i)9-s + (−0.630 + 0.0293i)10-s + (−0.0506 − 0.625i)11-s + (1.37 − 1.66i)12-s + (−0.543 − 0.0446i)13-s + (0.545 + 0.489i)14-s + (0.314 + 0.0235i)15-s + (1.04 + 1.45i)16-s + (0.0758 − 0.867i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-23.9i) \, \Gamma_{\R}(s-1.02i) \, \Gamma_{\R}(s+6.40i) \, \Gamma_{\R}(s+18.5i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.56270\) |
| Root analytic conductor: | \(1.11807\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((4,\ 1,\ (-23.9595040834i, -1.020656205786i, 6.40365485196i, 18.57650543722i:\ ),\ 1)\) |
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
−24.55979879, −23.90832344, −22.43266944, −16.89873756, −15.00588007, −12.88517163, −7.36091410, 3.21814688, 5.18369112, 9.27856928, 10.39178115, 11.43584182, 12.55999169, 14.57101229, 16.34814291, 18.60414862, 20.05552048, 20.94763417, 22.08917465