Dirichlet series
| L(s) = 1 | + (−0.342 + 0.686i)2-s + (−0.0731 − 0.234i)3-s + (−0.0111 + 0.216i)4-s + (−0.683 − 0.903i)5-s + (0.185 + 0.0300i)6-s + (−0.233 + 0.528i)7-s + (0.267 − 0.0817i)8-s + (0.0236 − 0.199i)9-s + (0.854 − 0.159i)10-s + (0.560 + 0.645i)11-s + (0.0514 − 0.0131i)12-s + (−0.383 − 1.14i)13-s + (−0.282 − 0.341i)14-s + (−0.161 + 0.226i)15-s + (−0.530 + 0.963i)16-s + (0.882 + 0.00446i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+32.0i) \, \Gamma_{\R}(s+9.89i) \, \Gamma_{\R}(s-41.9i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(53.5511\) |
| Root analytic conductor: | \(3.76926\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (32.0243446i, 9.89888556i, -41.9232302i:\ ),\ 1)\) |
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
−23.55665, −21.93718, −19.77986, −18.89877, −16.46109, −14.30409, −11.21655, −6.93323, −4.30776, −2.77544, −1.16072, 0.21313, 1.67819, 3.63572, 5.00656, 6.53565, 7.86422, 8.53643, 10.18105, 12.18268, 12.66076, 14.67509, 15.95216, 16.75683, 17.92749, 19.51726, 20.51436, 22.30503, 23.51780, 24.94767