Dirichlet series
| L(s) = 1 | + (0.337 − 0.202i)2-s + (−0.119 − 0.154i)3-s + (−0.264 − 0.338i)4-s + (−0.709 − 1.42i)5-s + (−0.0716 − 0.0278i)6-s + (−0.130 − 1.05i)7-s + (0.687 − 0.0608i)8-s + (0.110 − 0.117i)9-s + (−0.527 − 0.337i)10-s + (0.0779 − 0.177i)11-s + (−0.0205 + 0.0813i)12-s + (−1.13 + 1.61i)13-s + (−0.257 − 0.330i)14-s + (−0.134 + 0.280i)15-s + (0.577 − 0.193i)16-s + (−0.227 + 0.838i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+30.0i) \, \Gamma_{\R}(s+5.08i) \, \Gamma_{\R}(s-35.1i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(21.6402\) |
| Root analytic conductor: | \(2.78668\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (30.0701476i, 5.08628792i, -35.1564356i:\ ),\ 1)\) |
Euler product
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{3} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−22.73891, −22.15812, −19.47589, −17.95586, −15.71349, −14.47222, −12.31224, −10.43818, −7.61919, −2.86883, 0.31712, 1.50036, 4.23144, 4.68172, 6.97165, 8.49192, 10.10055, 11.81891, 13.01369, 14.20173, 16.27852, 17.05474, 19.43137, 20.04228, 21.81982, 23.63817, 24.04853