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
−24.24756559, −23.15714855, −21.31616502, −17.08215881, −14.10130838, −5.81915995, 2.54889254, 5.60457116, 9.49975839, 10.54545726, 11.77231603, 12.86634419, 14.70487035, 17.58801923, 19.01690025, 20.49198858, 21.61533006