Dirichlet series
| L(s) = 1 | + (0.185 + 2.22i)2-s + (−0.289 − 0.0268i)3-s + (−2.18 + 0.825i)4-s + (1.12 − 0.147i)5-s + (0.00628 − 0.649i)6-s + (0.185 − 0.780i)7-s + (−1.54 − 0.826i)8-s + (−0.587 + 0.0155i)9-s + (0.535 + 2.46i)10-s + (−0.104 − 0.674i)11-s + (0.653 − 0.180i)12-s + (−0.390 + 0.467i)13-s + (1.77 + 0.268i)14-s + (−0.328 + 0.0124i)15-s + (−0.435 − 1.33i)16-s + (0.875 − 0.239i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+22.8i) \, \Gamma_{\R}(s+1.48i) \, \Gamma_{\R}(s-7.63i) \, \Gamma_{\R}(s-16.6i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.63627\) |
| Root analytic conductor: | \(1.27422\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((4,\ 1,\ (22.8214852118i, 1.487955206482i, -7.6388645401i, -16.67057587822i:\ ),\ 1)\) |
Euler product
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.61533006, −20.49198858, −19.01690025, −17.58801923, −14.70487035, −12.86634419, −11.77231603, −10.54545726, −9.49975839, −5.60457116, −2.54889254, 5.81915995, 14.10130838, 17.08215881, 21.31616502, 23.15714855, 24.24756559