Dirichlet series
| L(s) = 1 | + 2.06·2-s + 1.79·3-s + 1.79·4-s + 0.454·5-s + 3.71·6-s + 0.0379·7-s + 0.675·8-s + 0.779·9-s + 0.939·10-s − 0.992·11-s + 3.23·12-s − 0.332·13-s + 0.0785·14-s + 0.816·15-s + 0.219·16-s − 0.298·17-s + 1.61·18-s + 0.400·19-s + 0.817·20-s + 0.0682·21-s − 2.05·22-s + 0.698·23-s + 1.21·24-s + 1.03·25-s − 0.687·26-s − 1.21·27-s + 0.0682·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+19.8i) \, \Gamma_{\R}(s+8.53i) \, \Gamma_{\R}(s-19.8i) \, \Gamma_{\R}(s-8.53i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(18.3179\) |
| Root analytic conductor: | \(2.06880\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 1,\ (19.81936708928i, 8.53107821814i, -19.81936708928i, -8.53107821814i:\ ),\ 1)\) |
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
−24.223274940, −22.660338142, −21.022531596, −14.944466160, −14.004231120, −13.074038138, −5.102901815, −3.371715172, −2.480449521, 2.480449521, 3.371715172, 5.102901815, 13.074038138, 14.004231120, 14.944466160, 21.022531596, 22.660338142, 24.223274940