Dirichlet series
| L(s) = 1 | + 0.605·2-s + 0.747·3-s + 0.369·4-s − 1.14·5-s + 0.453·6-s + 0.469·7-s + 0.831·8-s − 0.997·9-s − 0.692·10-s + 0.412·11-s + 0.276·12-s + 0.279·13-s + 0.284·14-s − 0.855·15-s − 0.128·16-s + 0.981·17-s − 0.604·18-s + 0.199·19-s − 0.422·20-s + 0.350·21-s + 0.249·22-s − 2.49·23-s + 0.620·24-s + 6.62·25-s + 0.160·26-s − 1.15·27-s + 0.100·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+16.1i) \, \Gamma_{\R}(s+3.71i) \, \Gamma_{\R}(s-16.1i) \, \Gamma_{\R}(s-3.71i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.29520\) |
| Root analytic conductor: | \(1.23085\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 1,\ (16.17300694712i, 3.71016260314i, -16.17300694712i, -3.71016260314i:\ ),\ 1)\) |
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
−23.570009434, −22.574065805, −20.543980006, −19.535544827, −14.237491804, −11.572828624, −8.087598018, 8.087598018, 11.572828624, 14.237491804, 19.535544827, 20.543980006, 22.574065805, 23.570009434