Dirichlet series
| L(s) = 1 | + (0.233 + 1.56i)2-s + (1.02 + 0.267i)3-s + (−0.709 + 0.728i)4-s + (−0.457 − 1.28i)5-s + (−0.178 + 1.66i)6-s + (−0.306 − 0.0628i)7-s + (−0.679 + 0.116i)8-s + (0.624 + 0.550i)9-s + (1.89 − 1.01i)10-s + (0.172 − 0.233i)11-s + (−0.924 + 0.559i)12-s + (−0.997 + 0.459i)13-s + (0.0266 − 0.493i)14-s + (−0.127 − 1.43i)15-s + (−0.0350 + 0.186i)16-s + (−0.196 + 0.0185i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-21.6i) \, \Gamma_{\R}(s-4.46i) \, \Gamma_{\R}(s+2.22i) \, \Gamma_{\R}(s+23.8i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.25166\) |
| Root analytic conductor: | \(1.34284\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((4,\ 1,\ (-21.6515998894i, -4.46755486328i, 2.22332030312i, 23.8958344496i:\ ),\ 1)\) |
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
−22.04354699, −20.19321749, −19.60274792, −18.40989391, −15.20440903, −14.14675135, −12.34948425, −11.04390092, −9.75538726, −7.28818408, −3.04158682, 7.49541914, 8.95555088, 12.68031797, 14.48040072, 15.94864304, 16.63990326, 19.87483120, 24.27912543, 24.86118025