Dirichlet series
| L(s) = 1 | + (1.04 + 0.375i)2-s + (−0.499 + 0.278i)3-s + (−0.0899 + 1.16i)4-s + (0.766 + 0.147i)5-s + (−0.627 + 0.105i)6-s + (0.337 − 0.150i)7-s + (−0.770 + 1.18i)8-s + (0.670 + 0.000535i)9-s + (0.748 + 0.441i)10-s + (0.527 − 0.341i)11-s + (−0.279 − 0.605i)12-s + (−0.270 − 0.253i)13-s + (0.410 − 0.0308i)14-s + (−0.423 + 0.140i)15-s + (−0.545 + 0.0759i)16-s + (0.223 + 0.180i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-13.5i) \, \Gamma_{\R}(s-4.76i) \, \Gamma_{\R}(s+18.3i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.78488\) |
| Root analytic conductor: | \(1.68509\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (-13.59658451496887i, -4.7646820672058325i, 18.361266582174704i:\ ),\ 1)\) |
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
−24.746399158399, −23.250147717264, −21.674353491122, −14.226820672061, −12.466771288438, −9.957369130168, −6.091232601062, −4.523743164535, −1.439862601725, 17.351331888344, 21.446121505329, 22.323920170806, 24.022422490338
Graph of the $Z$-function along the critical line
The lowest positive zero of this L-function, at height approximately 17.351, is higher than any other L-function of signature (0,0,0;).