Dirichlet series
L(s) = 1 | + 0.938·2-s + 0.906·3-s + 0.105·4-s + 1.59·5-s + 0.850·6-s − 0.756·7-s + 0.310·8-s − 0.422·9-s + 1.49·10-s − 0.142·11-s + 0.0957·12-s − 0.875·13-s − 0.709·14-s + 1.44·15-s + 0.0897·16-s − 0.566·17-s − 0.396·18-s + 0.916·19-s + 0.168·20-s − 0.685·21-s − 0.134·22-s − 0.148·23-s + 0.281·24-s + 1.10·25-s − 0.821·26-s − 0.604·27-s − 0.0798·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+17.3i) \, \Gamma_{\R}(s+5.55i) \, \Gamma_{\R}(s-17.3i) \, \Gamma_{\R}(s-5.55i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
Degree: | \(4\) |
Conductor: | \(1\) |
Sign: | $1$ |
Analytic conductor: | \(5.97805\) |
Root analytic conductor: | \(1.56365\) |
Rational: | no |
Arithmetic: | no |
Primitive: | yes |
Self-dual: | yes |
Selberg data: | \((4,\ 1,\ (17.38767874744i, 5.5596400005i, -17.38767874744i, -5.5596400005i:\ ),\ 1)\) |
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
−24.72274057, −22.71154679, −21.75768323, −20.08216834, −14.17824306, −13.26579338, −9.57307120, −2.52853940, 2.52853940, 9.57307120, 13.26579338, 14.17824306, 20.08216834, 21.75768323, 22.71154679, 24.72274057