Dirichlet series
| L(s) = 1 | + (−1.18 + 1.79i)2-s + (−0.553 − 0.202i)3-s + (−0.648 − 2.44i)4-s + (0.652 + 0.435i)5-s + (1.01 − 0.754i)6-s + (−0.0588 + 0.156i)7-s + (1.53 + 1.72i)8-s + (0.819 + 0.0217i)9-s + (−1.55 + 0.657i)10-s + (0.650 + 0.596i)11-s + (−0.136 + 1.48i)12-s + (0.0831 − 0.390i)13-s + (−0.210 − 0.290i)14-s + (−0.273 − 0.373i)15-s + (−2.47 − 1.53i)16-s + (−0.284 + 0.612i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-23.8i) \, \Gamma_{\R}(s-9.54i) \, \Gamma_{\R}(s+33.4i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(30.7144\) |
| Root analytic conductor: | \(3.13170\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (-23.880886064892i, -9.5487620156536i, 33.429648080546i:\ ),\ 1)\) |
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
−24.6543083456, −21.9453283034, −21.1571330731, −19.8784917065, −18.6461571285, −17.4706519447, −16.1526801250, −13.2268502248, −11.9341645645, −10.9007593470, −9.8177142906, −8.8754981553, −6.5745122068, −4.2480847859, −2.5489511134, −1.3008039372, −0.3775731732, 1.5530198958, 5.7643291691, 7.0471392490, 15.0737323938, 16.9618895628, 18.2114369131