Dirichlet series
| L(s) = 1 | + 0.409·2-s + 0.190·3-s + 0.368·4-s − 0.560·5-s + 0.0779·6-s + 1.32·7-s + 0.643·8-s + 0.222·9-s − 0.229·10-s + 1.63·11-s + 0.0701·12-s − 0.208·13-s + 0.542·14-s − 0.106·15-s − 0.494·16-s + 1.06·17-s + 0.0911·18-s − 0.712·19-s − 0.206·20-s + 0.251·21-s + 0.671·22-s + 0.0934·23-s + 0.122·24-s − 1.10·25-s − 0.0854·26-s + 0.267·27-s + 0.488·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+17.0i) \, \Gamma_{\R}(s+7.66i) \, \Gamma_{\R}(s-17.0i) \, \Gamma_{\R}(s-7.66i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(10.9006\) |
| Root analytic conductor: | \(1.81703\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 1,\ (17.02116595182i, 7.66420788368i, -17.02116595182i, -7.66420788368i:\ ),\ 1)\) |
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
−24.379481309, −23.021822302, −21.377053008, −19.748377233, −14.475707073, −11.613639475, −4.225630892, −1.563833957, 1.563833957, 4.225630892, 11.613639475, 14.475707073, 19.748377233, 21.377053008, 23.021822302, 24.379481309