Dirichlet series
| L(s) = 1 | + (0.273 + 0.465i)2-s + (0.329 + 0.248i)3-s + (−0.320 + 0.254i)4-s + (−0.346 + 0.489i)5-s + (−0.0252 + 0.221i)6-s + (0.322 + 0.106i)7-s + (0.0187 − 0.627i)8-s + (1.09 + 0.163i)9-s + (−0.322 − 0.0273i)10-s + (−0.408 + 0.172i)11-s + (−0.168 + 0.00451i)12-s + (−0.648 − 0.854i)13-s + (0.0387 + 0.179i)14-s + (−0.235 + 0.0753i)15-s + (−0.354 − 0.208i)16-s + (0.797 + 0.478i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+23.2i) \, \Gamma_{\R}(s+1.68i) \, \Gamma_{\R}(s-4.20i) \, \Gamma_{\R}(s-20.7i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.11699\) |
| Root analytic conductor: | \(1.20622\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((4,\ 1,\ (23.2225992068i, 1.68947169472i, -4.20024287272i, -20.7118280288i:\ ),\ 1)\) |
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
−24.28345749, −21.16057525, −19.67874877, −18.15528011, −16.15045305, −14.31422652, −12.96862181, −11.47918189, −9.49068159, −7.60572457, −4.66932844, 7.60308220, 10.14852158, 12.72455599, 14.63407008, 15.99288273, 18.40708320, 22.36013910, 24.18983025