L(s) = 1 | − 20·2-s + 120·4-s + 250·5-s − 100·7-s + 640·8-s − 5.00e3·10-s − 4.54e3·11-s + 3.54e3·13-s + 2.00e3·14-s − 1.15e4·16-s + 2.73e4·17-s + 3.87e4·19-s + 3.00e4·20-s + 9.08e4·22-s + 1.24e5·23-s + 4.68e4·25-s − 7.08e4·26-s − 1.20e4·28-s + 7.22e4·29-s + 3.06e5·31-s + 7.29e4·32-s − 5.46e5·34-s − 2.50e4·35-s − 1.23e5·37-s − 7.75e5·38-s + 1.60e5·40-s − 2.64e5·41-s + ⋯ |
L(s) = 1 | − 1.76·2-s + 0.937·4-s + 0.894·5-s − 0.110·7-s + 0.441·8-s − 1.58·10-s − 1.02·11-s + 0.446·13-s + 0.194·14-s − 0.707·16-s + 1.34·17-s + 1.29·19-s + 0.838·20-s + 1.81·22-s + 2.12·23-s + 3/5·25-s − 0.789·26-s − 0.103·28-s + 0.550·29-s + 1.84·31-s + 0.393·32-s − 2.38·34-s − 0.0985·35-s − 0.399·37-s − 2.29·38-s + 0.395·40-s − 0.599·41-s + ⋯ |
Λ(s)=(=(2025s/2ΓC(s)2L(s)Λ(8−s)
Λ(s)=(=(2025s/2ΓC(s+7/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
2025
= 34⋅52
|
Sign: |
1
|
Analytic conductor: |
197.608 |
Root analytic conductor: |
3.74931 |
Motivic weight: |
7 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 2025, ( :7/2,7/2), 1)
|
Particular Values
L(4) |
≈ |
1.225124744 |
L(21) |
≈ |
1.225124744 |
L(29) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 3 | | 1 |
| 5 | C1 | (1−p3T)2 |
good | 2 | D4 | 1+5p2T+35p3T2+5p9T3+p14T4 |
| 7 | D4 | 1+100T+1411250T2+100p7T3+p14T4 |
| 11 | D4 | 1+4544T+31976326T2+4544p7T3+p14T4 |
| 13 | D4 | 1−3540T+100535470T2−3540p7T3+p14T4 |
| 17 | D4 | 1−27340T+901005190T2−27340p7T3+p14T4 |
| 19 | D4 | 1−2040pT+113449762pT2−2040p8T3+p14T4 |
| 23 | D4 | 1−124140T+10649684530T2−124140p7T3+p14T4 |
| 29 | D4 | 1−72260T+6846819118T2−72260p7T3+p14T4 |
| 31 | D4 | 1−306824T+77964629966T2−306824p7T3+p14T4 |
| 37 | D4 | 1+123020T+144088599870T2+123020p7T3+p14T4 |
| 41 | D4 | 1+264364T+161786388886T2+264364p7T3+p14T4 |
| 43 | D4 | 1−423300T+446651231050T2−423300p7T3+p14T4 |
| 47 | D4 | 1−105460T+858715356610T2−105460p7T3+p14T4 |
| 53 | D4 | 1−2391580T+3562552504510T2−2391580p7T3+p14T4 |
| 59 | D4 | 1−1120120T+1362334883638T2−1120120p7T3+p14T4 |
| 61 | D4 | 1−2257044T+5613447576526T2−2257044p7T3+p14T4 |
| 67 | D4 | 1−4516460T+16742087664890T2−4516460p7T3+p14T4 |
| 71 | D4 | 1+621784T+17914494152446T2+621784p7T3+p14T4 |
| 73 | D4 | 1−4569060T+23424949855030T2−4569060p7T3+p14T4 |
| 79 | D4 | 1−4333040T+330830231042pT2−4333040p7T3+p14T4 |
| 83 | D4 | 1−9793020T+59971104320890T2−9793020p7T3+p14T4 |
| 89 | D4 | 1+6025620T+89865866149558T2+6025620p7T3+p14T4 |
| 97 | D4 | 1−4609540T+142930351581510T2−4609540p7T3+p14T4 |
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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
−14.59852360063796427733243724814, −14.00493737382901278011603018360, −13.39297133610143059644540145269, −13.03548884604674544987808746051, −12.15185124406984080145182921076, −11.45922302357553324708101202474, −10.45154286020594823189006120946, −10.37660375167141559171435437691, −9.507247035329806785202718061160, −9.416490350687852440185497505153, −8.369841317355816189714843271932, −8.163443991431038144273715191075, −7.27052843578853919982484826210, −6.53761132291264760759620629903, −5.37665655989887589678139222999, −5.04126837951953862533885090518, −3.37492276241368616820483871399, −2.50133437303367066785562798812, −0.970534731658867784971968568513, −0.896116664342765117196953890616,
0.896116664342765117196953890616, 0.970534731658867784971968568513, 2.50133437303367066785562798812, 3.37492276241368616820483871399, 5.04126837951953862533885090518, 5.37665655989887589678139222999, 6.53761132291264760759620629903, 7.27052843578853919982484826210, 8.163443991431038144273715191075, 8.369841317355816189714843271932, 9.416490350687852440185497505153, 9.507247035329806785202718061160, 10.37660375167141559171435437691, 10.45154286020594823189006120946, 11.45922302357553324708101202474, 12.15185124406984080145182921076, 13.03548884604674544987808746051, 13.39297133610143059644540145269, 14.00493737382901278011603018360, 14.59852360063796427733243724814