Dirichlet series
| L(s) = 1 | + 0.736·2-s + 0.998·3-s − 0.919·4-s + 0.267·5-s + 0.735·6-s − 0.0310·7-s − 1.01·8-s + 0.519·9-s + 0.197·10-s + 0.313·11-s − 0.918·12-s + 0.547·13-s − 0.0229·14-s + 0.267·15-s + 0.137·16-s − 1.20·17-s + 0.382·18-s + 0.734·19-s − 0.246·20-s − 0.0310·21-s + 0.231·22-s + 0.330·23-s − 1.01·24-s − 0.592·25-s + 0.403·26-s + 1.04·27-s + 0.0200·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+14.7i) \, \Gamma_{\R}(s+4.02i) \, \Gamma_{\R}(s-14.7i) \, \Gamma_{\R}(s-4.02i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.24328\) |
| Root analytic conductor: | \(1.22382\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 1,\ (14.7254797045i, 4.02669357914i, -14.7254797045i, -4.02669357914i:\ ),\ 1)\) |
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
−24.09272184, −22.70577799, −21.63672770, −20.04510826, −18.20991974, −13.47382769, −9.01112667, 9.01112667, 13.47382769, 18.20991974, 20.04510826, 21.63672770, 22.70577799, 24.09272184