Dirichlet series
| L(s) = 1 | + (0.0546 + 0.376i)2-s + (−1.01 − 0.326i)3-s + (0.118 + 0.0411i)4-s + (−0.484 + 0.357i)5-s + (0.0675 − 0.398i)6-s + (0.0765 + 0.454i)7-s + (0.0597 − 0.232i)8-s + (−0.353 + 0.660i)9-s + (−0.160 − 0.162i)10-s + (1.51 + 0.525i)11-s + (−0.106 − 0.0803i)12-s + (−0.158 − 1.19i)13-s + (−0.166 + 0.0536i)14-s + (0.606 − 0.203i)15-s + (−0.734 + 0.0203i)16-s + (−0.563 + 0.0960i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+21.5i) \, \Gamma_{\R}(s+0.411i) \, \Gamma_{\R}(s-6.41i) \, \Gamma_{\R}(s-15.4i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.206787\) |
| Root analytic conductor: | \(0.674343\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((4,\ 1,\ (21.5036539084i, 0.411557862658i, -6.41813484886i, -15.49707692222i:\ ),\ 1)\) |
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
−24.39307003, −22.87442106, −19.77964434, −17.34849621, −16.27684931, −14.14540549, −11.76043085, −11.29039870, −9.06012294, −6.47550246, −4.36507219, 11.67444402, 17.62750278, 19.76575454, 22.38641290, 22.83540371, 24.57370914