Dirichlet series
| L(s) = 1 | + 0.399·2-s + 0.691·3-s + 0.188·4-s + 0.779·5-s + 0.275·6-s + 1.41·7-s + 0.485·8-s − 1.12·9-s + 0.311·10-s − 0.147·11-s + 0.130·12-s + 0.399·13-s + 0.565·14-s + 0.539·15-s − 0.641·16-s + 0.241·17-s − 0.448·18-s − 1.46·19-s + 0.146·20-s + 0.980·21-s − 0.0589·22-s + 1.09·23-s + 0.335·24-s − 0.780·25-s + 0.159·26-s − 1.19·27-s + 0.266·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+19.9i) \, \Gamma_{\R}(s+5.19i) \, \Gamma_{\R}(s-19.9i) \, \Gamma_{\R}(s-5.19i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.87559\) |
| Root analytic conductor: | \(1.61930\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 1,\ (19.9527202619i, 5.19676055138i, -19.9527202619i, -5.19676055138i:\ ),\ 1)\) |
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
−23.26929688, −21.13871942, −17.37032184, −14.70281067, −13.64422636, −11.12624444, −8.52564450, −2.09458321, 2.09458321, 8.52564450, 11.12624444, 13.64422636, 14.70281067, 17.37032184, 21.13871942, 23.26929688