Dirichlet series
| L(s) = 1 | + 0.135·2-s + 0.458·3-s + 0.673·4-s + 0.109·5-s + 0.0620·6-s − 0.918·7-s + 0.314·8-s − 0.485·9-s + 0.0147·10-s + 1.07·11-s + 0.308·12-s − 0.0300·13-s − 0.124·14-s + 0.0500·15-s − 0.498·16-s − 0.697·17-s − 0.0656·18-s − 0.0725·19-s + 0.0734·20-s − 0.421·21-s + 0.145·22-s + 2.64·23-s + 0.144·24-s − 1.13·25-s − 0.00400·26-s − 0.0830·27-s − 0.617·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+20.7i) \, \Gamma_{\R}(s+2.90i) \, \Gamma_{\R}(s-20.7i) \, \Gamma_{\R}(s-2.90i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.30957\) |
| Root analytic conductor: | \(1.23277\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 1,\ (20.7698952908i, 2.9045205178i, -20.7698952908i, -2.9045205178i:\ ),\ 1)\) |
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
−24.86773927, −22.74241717, −19.54248225, −16.81940639, −15.11984282, −13.33323885, −11.35949249, −9.19320186, −6.72766930, 6.72766930, 9.19320186, 11.35949249, 13.33323885, 15.11984282, 16.81940639, 19.54248225, 22.74241717, 24.86773927