Dirichlet series
| L(s) = 1 | + (−0.694 + 1.13i)2-s + (0.0926 + 0.763i)3-s + (0.0844 − 1.58i)4-s + (−0.0907 + 0.121i)5-s + (−0.934 − 0.425i)6-s + (−0.638 + 0.00635i)7-s + (0.425 + 1.08i)8-s + (0.473 + 0.141i)9-s + (−0.0756 − 0.187i)10-s + (−0.550 + 0.285i)11-s + (1.21 − 0.0820i)12-s + (0.120 + 0.398i)13-s + (0.436 − 0.732i)14-s + (−0.101 − 0.0580i)15-s + (−0.671 − 1.69i)16-s + (0.00521 + 0.847i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+22.0i) \, \Gamma_{\R}(s+0.793i) \, \Gamma_{\R}(s-6.49i) \, \Gamma_{\R}(s-16.3i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.847099\) |
| Root analytic conductor: | \(0.959364\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((4,\ 1,\ (22.070791094i, 0.793600760768i, -6.49925364748i, -16.36513820726i:\ ),\ 1)\) |
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
−24.5203059, −20.6340103, −19.0819757, −17.8281703, −15.9389955, −13.0389393, −12.2041366, −10.6338184, −8.7457703, −7.0330454, −2.9235541, 9.9367614, 15.3778623, 19.0265308, 21.1640163, 23.2541149, 24.4561235