Dirichlet series
| L(s) = 1 | + (−0.473 − 0.345i)2-s + (0.902 − 0.0126i)3-s + (−0.525 + 0.327i)4-s + (−0.0303 + 0.847i)5-s + (−0.431 − 0.306i)6-s + (−0.348 + 0.611i)7-s + (0.187 + 0.590i)8-s + (0.799 − 0.0227i)9-s + (0.307 − 0.391i)10-s + (0.256 − 0.745i)11-s + (−0.470 + 0.302i)12-s + (−0.477 − 0.472i)13-s + (0.376 − 0.169i)14-s + (−0.0167 + 0.765i)15-s + (−0.209 − 0.551i)16-s + (0.205 + 0.140i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+22.5i) \, \Gamma_{\R}(s+0.877i) \, \Gamma_{\R}(s-8.89i) \, \Gamma_{\R}(s-14.5i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.25130\) |
| Root analytic conductor: | \(1.05764\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((4,\ 1,\ (22.5224000508i, 0.877793680384i, -8.89271031532i, -14.50748341588i:\ ),\ 1)\) |
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
−24.40107476, −20.35447110, −18.94507322, −17.17431380, −15.62993783, −13.96602377, −12.66475243, −9.81023223, −8.92106206, −7.21584682, −4.36082312, 2.90315745, 18.29701111, 19.52958908, 21.38370873, 22.71078844, 24.67850357