Dirichlet series
| L(s) = 1 | + (−0.473 + 0.792i)2-s + (0.125 + 0.586i)3-s + (0.0692 + 0.0426i)4-s + (−0.337 − 0.494i)5-s + (−0.523 − 0.178i)6-s + (−0.876 − 1.30i)7-s + (0.0819 + 0.0346i)8-s + (−0.453 + 0.733i)9-s + (0.551 − 0.0329i)10-s + (0.0269 − 0.224i)11-s + (−0.0163 + 0.0459i)12-s + (0.641 + 0.578i)13-s + (1.45 − 0.0763i)14-s + (0.247 − 0.259i)15-s + (−0.540 + 0.915i)16-s + (−1.00 − 0.659i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+22.7i) \, \Gamma_{\R}(s+0.284i) \, \Gamma_{\R}(s-23.0i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.166709\) |
| Root analytic conductor: | \(0.550368\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (22.76448752754i, 0.28490754560446i, -23.049395073144i:\ ),\ 1)\) |
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
−19.650154863, −18.363523023, −15.293539603, −12.680085224, −10.999878555, −8.993535098, −6.492101737, −2.652598497, 4.142238115, 6.926557300, 8.608671976, 10.692953299, 13.508498379, 16.076453033, 16.763938763, 19.926736154