Dirichlet series
| L(s) = 1 | + (0.701 − 0.611i)2-s + (0.309 + 0.662i)3-s + (−0.583 − 1.46i)4-s + (0.775 − 0.0519i)5-s + (0.621 + 0.275i)6-s + (−0.843 + 0.0894i)7-s + (−1.17 − 0.673i)8-s + (−0.652 + 1.07i)9-s + (0.511 − 0.510i)10-s + (−0.718 − 1.27i)11-s + (0.793 − 0.840i)12-s + (−0.413 + 0.173i)13-s + (−0.537 + 0.578i)14-s + (0.274 + 0.497i)15-s + (−1.02 + 1.01i)16-s + (−0.377 + 0.443i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+32.8i) \, \Gamma_{\R}(s+4.93i) \, \Gamma_{\R}(s-37.7i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(24.6204\) |
| Root analytic conductor: | \(2.90914\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (32.8171198i, 4.93784756i, -37.7549674i:\ ),\ 1)\) |
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
−23.41470, −22.24220, −20.53757, −18.10870, −17.06184, −14.86698, −13.25552, −12.61864, −9.18329, −7.11526, −2.74215, 0.14220, 2.11578, 3.53904, 5.11313, 5.93683, 8.81804, 10.00169, 10.85109, 13.17172, 13.77441, 15.08019, 16.66448, 18.80126, 19.64371, 21.38737, 22.11932, 23.40527