L(s) = 1 | − 36·5-s − 120·7-s + 200·11-s + 284·13-s − 2.67e3·17-s + 72·19-s − 3.84e3·23-s + 2.65e3·25-s − 1.02e4·29-s + 1.04e4·31-s + 4.32e3·35-s + 1.31e4·37-s − 4.16e3·41-s − 5.83e3·43-s − 1.52e3·47-s − 1.48e4·49-s − 9.01e3·53-s − 7.20e3·55-s + 5.50e4·59-s − 6.34e4·61-s − 1.02e4·65-s + 3.67e4·67-s − 3.76e4·71-s − 3.78e4·73-s − 2.40e4·77-s + 1.44e5·79-s + 1.09e5·83-s + ⋯ |
L(s) = 1 | − 0.643·5-s − 0.925·7-s + 0.498·11-s + 0.466·13-s − 2.24·17-s + 0.0457·19-s − 1.51·23-s + 0.850·25-s − 2.25·29-s + 1.96·31-s + 0.596·35-s + 1.57·37-s − 0.386·41-s − 0.481·43-s − 0.100·47-s − 0.885·49-s − 0.440·53-s − 0.320·55-s + 2.06·59-s − 2.18·61-s − 0.300·65-s + 1.00·67-s − 0.886·71-s − 0.830·73-s − 0.461·77-s + 2.61·79-s + 1.74·83-s + ⋯ |
Λ(s)=(=(82944s/2ΓC(s)2L(s)Λ(6−s)
Λ(s)=(=(82944s/2ΓC(s+5/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
82944
= 210⋅34
|
Sign: |
1
|
Analytic conductor: |
2133.56 |
Root analytic conductor: |
6.79636 |
Motivic weight: |
5 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 82944, ( :5/2,5/2), 1)
|
Particular Values
L(3) |
≈ |
0.2590648598 |
L(21) |
≈ |
0.2590648598 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 3 | | 1 |
good | 5 | D4 | 1+36T−1362T2+36p5T3+p10T4 |
| 7 | D4 | 1+120T+29278T2+120p5T3+p10T4 |
| 11 | D4 | 1−200T+46406T2−200p5T3+p10T4 |
| 13 | D4 | 1−284T+477054T2−284p5T3+p10T4 |
| 17 | D4 | 1+2676T+4344262T2+2676p5T3+p10T4 |
| 19 | D4 | 1−72T+4159894T2−72p5T3+p10T4 |
| 23 | D4 | 1+3840T+9416686T2+3840p5T3+p10T4 |
| 29 | D4 | 1+10212T+64228638T2+10212p5T3+p10T4 |
| 31 | D4 | 1−10488T+84559438T2−10488p5T3+p10T4 |
| 37 | D4 | 1−13148T+125908974T2−13148p5T3+p10T4 |
| 41 | D4 | 1+4164T+216207126T2+4164p5T3+p10T4 |
| 43 | D4 | 1+5832T+168401542T2+5832p5T3+p10T4 |
| 47 | D4 | 1+1520T+148144670T2+1520p5T3+p10T4 |
| 53 | D4 | 1+9012T+816689646T2+9012p5T3+p10T4 |
| 59 | D4 | 1−55096T+1818478886T2−55096p5T3+p10T4 |
| 61 | D4 | 1+63444T+2677193342T2+63444p5T3+p10T4 |
| 67 | D4 | 1−36792T+1148752246T2−36792p5T3+p10T4 |
| 71 | D4 | 1+37664T+2440628942T2+37664p5T3+p10T4 |
| 73 | D4 | 1+37836T+2085902966T2+37836p5T3+p10T4 |
| 79 | D4 | 1−144888T+10711615534T2−144888p5T3+p10T4 |
| 83 | D4 | 1−109272T+10472055958T2−109272p5T3+p10T4 |
| 89 | D4 | 1−32556T+8836559958T2−32556p5T3+p10T4 |
| 97 | D4 | 1−69092T+18339537030T2−69092p5T3+p10T4 |
show more | | |
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L(s)=p∏ j=1∏4(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−11.10349898750354388844759098873, −10.99592506406160479889289097081, −10.15761506248492035931915018225, −9.813597982188160246565459516624, −9.207721128276852356687927329682, −8.951303769847119867169886672344, −8.304242936937685415001827320874, −7.88489096029691078771304770369, −7.32347867296518986850370489202, −6.62992318519195227892801964667, −6.25151844223162851296684755208, −6.10173524444721992750269603546, −4.91926282841069563636886183455, −4.57694241511532075752866878131, −3.71737995047683625391745563519, −3.67160078121328846309029964919, −2.59405891947027587264839096536, −2.09299084263045959209608581651, −1.11808101953705506783798796545, −0.15018176816877938014164731810,
0.15018176816877938014164731810, 1.11808101953705506783798796545, 2.09299084263045959209608581651, 2.59405891947027587264839096536, 3.67160078121328846309029964919, 3.71737995047683625391745563519, 4.57694241511532075752866878131, 4.91926282841069563636886183455, 6.10173524444721992750269603546, 6.25151844223162851296684755208, 6.62992318519195227892801964667, 7.32347867296518986850370489202, 7.88489096029691078771304770369, 8.304242936937685415001827320874, 8.951303769847119867169886672344, 9.207721128276852356687927329682, 9.813597982188160246565459516624, 10.15761506248492035931915018225, 10.99592506406160479889289097081, 11.10349898750354388844759098873