Dirichlet series
| L(s) = 1 | + (0.0281 − 1.02i)2-s + (0.578 − 1.12i)3-s + (−0.251 − 0.0577i)4-s + (−0.651 + 0.870i)5-s + (−1.13 − 0.624i)6-s + (0.198 + 0.0372i)7-s + (−0.0156 + 0.462i)8-s + (−0.690 − 1.30i)9-s + (0.874 + 0.692i)10-s + (0.278 − 0.420i)11-s + (−0.210 + 0.249i)12-s + (0.339 − 0.264i)13-s + (0.0437 − 0.202i)14-s + (0.602 + 1.23i)15-s + (0.324 − 0.0170i)16-s + (−0.506 + 0.148i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-22.3i) \, \Gamma_{\R}(s-1.34i) \, \Gamma_{\R}(s+8.50i) \, \Gamma_{\R}(s+15.2i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.31581\) |
| Root analytic conductor: | \(1.23360\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((4,\ 1,\ (-22.3834989096i, -1.340802471452i, 8.50955223286i, 15.2147491482i:\ ),\ 1)\) |
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
−24.68361868, −23.04239701, −21.31149407, −19.87500283, −16.28621256, −4.35458603, 3.01345575, 6.85474727, 8.52758104, 10.99605834, 11.95224794, 13.60742476, 15.02143087, 17.90152081, 19.17333983, 20.23587574, 23.90013186