Dirichlet series
| L(s) = 1 | + (0.141 + 0.853i)2-s + (0.0379 + 0.413i)3-s + (−0.850 + 1.09i)4-s + (−0.0982 − 0.668i)5-s + (−0.347 + 0.0910i)6-s + (−0.490 − 0.998i)7-s + (−0.804 − 0.570i)8-s + (−0.207 + 0.444i)9-s + (0.557 − 0.178i)10-s + (0.183 − 0.858i)11-s + (−0.485 − 0.310i)12-s + (1.62 − 0.191i)13-s + (0.783 − 0.559i)14-s + (0.272 − 0.0660i)15-s + (−0.299 − 0.795i)16-s + (−0.569 + 0.201i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+32.9i) \, \Gamma_{\R}(s+3.50i) \, \Gamma_{\R}(s-36.4i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(16.9237\) |
| Root analytic conductor: | \(2.56743\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (32.987429682i, 3.5002128372i, -36.48764252i:\ ),\ 1)\) |
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
−23.420472, −22.423698, −20.432673, −18.868431, −18.266187, −15.318678, −13.923596, −12.247686, −10.658646, −9.088982, −6.163996, −1.593869, 0.455495, 3.569462, 4.514330, 6.397202, 8.030527, 8.954849, 10.957415, 13.085643, 13.697314, 15.975519, 16.475513, 17.725395, 19.832215, 21.380980, 22.632144, 23.996557