Dirichlet series
| L(s) = 1 | + (−0.250 − 0.433i)2-s + (0.360 + 0.356i)3-s + (−0.125 + 0.216i)4-s + (−0.592 + 0.583i)5-s + (0.0642 − 0.245i)6-s + (0.103 + 1.02i)7-s +(0.125)·8-s + (−0.357 + 0.613i)9-s + (0.400 + 0.110i)10-s + (1.34 + 0.122i)11-s + (−0.122 + 0.0334i)12-s + (0.0616 − 0.416i)13-s + (0.418 − 0.301i)14-s + (−0.421 − 0.000934i)15-s + (−0.0312 − 0.0541i)16-s + (−0.855 − 0.598i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\R}(s-8.95i) \, \Gamma_{\R}(s-2.93i) \, \Gamma_{\R}(s+11.8i) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(4\) = \(2^{2}\) |
| Sign: | $-0.5 + 0.866i$ |
| Analytic conductor: | \(5.01196\) |
| Root analytic conductor: | \(1.71133\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 4,\ (-8.9546625172i, -2.9365915306i, 11.8912540478i:\ ),\ -0.5 + 0.866i)\) |
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
−24.35540600, −23.85482092, −22.34994240, −20.25256015, −19.64960226, −17.69700217, −16.34014697, −14.26571491, −8.72396898, −6.85564032, −4.07774542, −0.93372346, 11.70886651, 14.40713480, 15.92737505, 17.95238412, 19.31006000, 20.29133986, 22.01745593, 22.57913504, 24.59912256