Dirichlet series
| L(s) = 1 | − 1.10·2-s − 1.55·3-s + 0.437·4-s − 1.36·5-s + 1.72·6-s + 0.969·7-s − 0.714·8-s + 0.279·9-s + 1.51·10-s − 0.870·11-s − 0.678·12-s + 0.488·13-s − 1.07·14-s + 2.12·15-s + 0.676·16-s − 0.451·17-s − 0.309·18-s − 0.119·19-s − 0.598·20-s − 1.50·21-s + 0.965·22-s + 0.00158·23-s + 1.10·24-s + 0.160·25-s − 0.541·26-s + 1.32·27-s + 0.423·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+20.5i) \, \Gamma_{\R}(s+5.41i) \, \Gamma_{\R}(s-20.5i) \, \Gamma_{\R}(s-5.41i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.94181\) |
| Root analytic conductor: | \(1.67872\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 1,\ (20.5888628182i, 5.411923388i, -20.5888628182i, -5.411923388i:\ ),\ 1)\) |
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
−24.0361519, −23.0949784, −18.3385992, −17.1932871, −15.5399923, −11.8080457, −11.0295618, −8.2511210, −0.3674000, 0.3674000, 8.2511210, 11.0295618, 11.8080457, 15.5399923, 17.1932871, 18.3385992, 23.0949784, 24.0361519