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
−24.67850357, −22.71078844, −21.38370873, −19.52958908, −18.29701111, −2.90315745, 4.36082312, 7.21584682, 8.92106206, 9.81023223, 12.66475243, 13.96602377, 15.62993783, 17.17431380, 18.94507322, 20.35447110, 24.40107476