Dirichlet series
| L(s) = 1 | + (0.552 − 0.674i)2-s + (−0.645 + 0.802i)3-s + (−0.702 − 1.42i)4-s + (−0.212 + 0.838i)5-s + (0.184 + 0.878i)6-s + (−0.589 − 0.225i)7-s + (−1.10 − 0.311i)8-s + (0.418 − 0.233i)9-s + (0.448 + 0.606i)10-s + (−0.258 − 0.604i)11-s + (1.59 + 0.353i)12-s + (0.962 − 0.349i)13-s + (−0.477 + 0.273i)14-s + (−0.535 − 0.711i)15-s + (−0.839 + 1.15i)16-s + (−0.457 − 0.615i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-21.8i) \, \Gamma_{\R}(s-0.112i) \, \Gamma_{\R}(s+22.0i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.0689440\) |
| Root analytic conductor: | \(0.410045\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (-21.8973727191888i, -0.112784630517544i, 22.0101573497062i:\ ),\ 1)\) |
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
−23.79573742910, −17.82510420835, −16.16834532701, −13.13159420354, −12.37540939066, −8.50360469347, −6.61925624517, −4.42635973003, 3.95448897162, 5.85609407616, 9.99612127372, 10.98035332675, 13.61218893684, 15.59466648955, 18.79673717673, 23.01483367112