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
−22.08917465, −20.94763417, −20.05552048, −18.60414862, −16.34814291, −14.57101229, −12.55999169, −11.43584182, −10.39178115, −9.27856928, −5.18369112, −3.21814688, 7.36091410, 12.88517163, 15.00588007, 16.89873756, 22.43266944, 23.90832344, 24.55979879