Dirichlet series
| L(s) = 1 | + (−0.191 − 0.587i)2-s + (−0.652 + 0.963i)3-s + (−0.530 + 0.224i)4-s + (−0.312 − 0.0112i)5-s + (0.690 + 0.199i)6-s + (−0.442 + 0.00815i)7-s + (0.0848 + 0.987i)8-s + (−0.306 − 1.25i)9-s + (0.0531 + 0.185i)10-s + (0.346 + 0.688i)11-s + (0.130 − 0.658i)12-s + (−0.501 + 0.280i)13-s + (0.0894 + 0.258i)14-s + (0.214 − 0.293i)15-s + (0.0634 − 0.288i)16-s + (0.0617 + 0.240i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+13.0i) \, \Gamma_{\R}(s+6.72i) \, \Gamma_{\R}(s-4.36i) \, \Gamma_{\R}(s-15.4i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.77594\) |
| Root analytic conductor: | \(1.39397\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((4,\ 1,\ (13.04820238558i, 6.72621058578i, -4.36776322526i, -15.4066497461i:\ ),\ 1)\) |
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
−24.46070856, −23.07678971, −22.13190107, −19.41945021, −18.29074034, −16.48297903, −0.37474707, 9.59674098, 11.68799158, 17.54386221, 19.86792005, 21.31486727, 22.57774563, 23.55832145