Dirichlet series
| L(s) = 1 | − 1.58·2-s − 0.484·3-s + 1.06·4-s + 0.265·5-s + 0.766·6-s − 0.488·7-s − 0.979·8-s − 1.67·9-s − 0.420·10-s − 1.01·11-s − 0.515·12-s + 0.0308·13-s + 0.773·14-s − 0.128·15-s + 1.52·16-s − 1.80·17-s + 2.64·18-s − 0.164·19-s + 0.282·20-s + 0.236·21-s + 1.61·22-s − 2.26·23-s + 0.473·24-s + 1.39·25-s − 0.0488·26-s + 1.24·27-s − 0.519·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+16.8i) \, \Gamma_{\R}(s+10.7i) \, \Gamma_{\R}(s-16.8i) \, \Gamma_{\R}(s-10.7i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(20.9147\) |
| Root analytic conductor: | \(2.13851\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 1,\ (16.82022526612i, 10.73927743766i, -16.82022526612i, -10.73927743766i:\ ),\ 1)\) |
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
−24.130866499, −22.398824839, −20.117036561, −18.049903594, −8.631719229, −5.938198592, −2.550771401, −0.142749235, 0.142749235, 2.550771401, 5.938198592, 8.631719229, 18.049903594, 20.117036561, 22.398824839, 24.130866499