Dirichlet series
L(s) = 1 | + (0.979 − 0.201i)2-s + (0.979 + 0.201i)3-s + (0.918 − 0.394i)4-s + (−0.0506 + 0.998i)5-s + 6-s + (−0.758 − 0.651i)7-s + (0.820 − 0.571i)8-s + (0.918 + 0.394i)9-s + (0.151 + 0.988i)10-s + (0.994 − 0.101i)11-s + (0.979 − 0.201i)12-s + (0.918 + 0.394i)13-s + (−0.874 − 0.485i)14-s + (−0.250 + 0.968i)15-s + (0.688 − 0.724i)16-s + (−0.151 − 0.988i)17-s + ⋯ |
L(s) = 1 | + (0.979 − 0.201i)2-s + (0.979 + 0.201i)3-s + (0.918 − 0.394i)4-s + (−0.0506 + 0.998i)5-s + 6-s + (−0.758 − 0.651i)7-s + (0.820 − 0.571i)8-s + (0.918 + 0.394i)9-s + (0.151 + 0.988i)10-s + (0.994 − 0.101i)11-s + (0.979 − 0.201i)12-s + (0.918 + 0.394i)13-s + (−0.874 − 0.485i)14-s + (−0.250 + 0.968i)15-s + (0.688 − 0.724i)16-s + (−0.151 − 0.988i)17-s + ⋯ |
Functional equation
Invariants
Degree: | \(1\) |
Conductor: | \(311\) |
Sign: | $0.983 + 0.183i$ |
Analytic conductor: | \(33.4215\) |
Root analytic conductor: | \(33.4215\) |
Motivic weight: | \(0\) |
Rational: | no |
Arithmetic: | yes |
Character: | $\chi_{311} (171, \cdot )$ |
Primitive: | yes |
Self-dual: | no |
Analytic rank: | \(0\) |
Selberg data: | \((1,\ 311,\ (1:\ ),\ 0.983 + 0.183i)\) |
Particular Values
\(L(\frac{1}{2})\) | \(\approx\) | \(5.607527770 + 0.5179849215i\) |
\(L(\frac12)\) | \(\approx\) | \(5.607527770 + 0.5179849215i\) |
\(L(1)\) | \(\approx\) | \(2.788798567 + 0.09500446186i\) |
\(L(1)\) | \(\approx\) | \(2.788798567 + 0.09500446186i\) |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 311 | \( 1 \) |
good | 2 | \( 1 + (0.979 - 0.201i)T \) |
3 | \( 1 + (0.979 + 0.201i)T \) | |
5 | \( 1 + (-0.0506 + 0.998i)T \) | |
7 | \( 1 + (-0.758 - 0.651i)T \) | |
11 | \( 1 + (0.994 - 0.101i)T \) | |
13 | \( 1 + (0.918 + 0.394i)T \) | |
17 | \( 1 + (-0.151 - 0.988i)T \) | |
19 | \( 1 + (0.0506 + 0.998i)T \) | |
23 | \( 1 + (-0.820 + 0.571i)T \) | |
29 | \( 1 + (0.874 - 0.485i)T \) | |
31 | \( 1 + (-0.151 + 0.988i)T \) | |
37 | \( 1 + (0.758 - 0.651i)T \) | |
41 | \( 1 + (0.440 - 0.897i)T \) | |
43 | \( 1 + (0.758 + 0.651i)T \) | |
47 | \( 1 + (-0.874 + 0.485i)T \) | |
53 | \( 1 + (-0.758 - 0.651i)T \) | |
59 | \( 1 + (0.758 + 0.651i)T \) | |
61 | \( 1 + (0.0506 + 0.998i)T \) | |
67 | \( 1 + (-0.440 - 0.897i)T \) | |
71 | \( 1 + (-0.528 - 0.848i)T \) | |
73 | \( 1 + (0.347 - 0.937i)T \) | |
79 | \( 1 + (-0.440 + 0.897i)T \) | |
83 | \( 1 + (-0.250 - 0.968i)T \) | |
89 | \( 1 + (-0.758 + 0.651i)T \) | |
97 | \( 1 + (-0.528 - 0.848i)T \) | |
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Imaginary part of the first few zeros on the critical line
−24.9530730012816838124434485957, −24.20591755550069151849325149675, −23.415745348937179824426755570858, −22.11706596794014106619869849474, −21.54955531620616348952149016044, −20.386464454708427057101932761523, −19.94607426721238108537616062912, −19.0852085931239408886073499819, −17.63768082691550464060080186360, −16.419318589844120770115130412891, −15.695214928414823438978739015590, −14.94738465258773459192639857887, −13.87255619141807892904457509719, −12.97092853223776666126856425924, −12.567348683248152638351762124150, −11.46368429337648428135300878496, −9.85559112781702545406328693824, −8.763611197880579391321580282347, −8.13118423606830803640155256143, −6.675981972648792308265054786509, −5.92832114126941447820811826010, −4.44909347573031722112597005457, −3.67732989863936338208579570380, −2.51118822549346829581428977808, −1.28057492687278021114310209425, 1.44982077899366882962578156185, 2.77857434973957359112409375404, 3.637529606444848900306395171932, 4.20652447265654468823044180445, 6.077777046115065929350346103458, 6.84410598415771426047602457589, 7.74916167006874797892626384774, 9.37187576578425045570828136685, 10.196330994990650060212008975, 11.136637707805663722942885340331, 12.226664856844395263883855532936, 13.51400204445971113742512694829, 14.03708668860305890787586452406, 14.60884207989977962127153424730, 15.88509971483886576933998397330, 16.2641871812411074040489967361, 18.072347371411511616441299493356, 19.31840110513395775618599115612, 19.57942800710578368821431185262, 20.6832277967293992862651165414, 21.46065808569504216909069278773, 22.48744420960844845996608144801, 22.96505467474005126227078488999, 24.068144372561431720262900562283, 25.29530646892760118450507084470