L(s) = 1 | + 15·5-s − 14·7-s − 22·11-s + 8·13-s + 34·17-s + 4·19-s − 176·23-s + 150·25-s − 98·29-s − 88·31-s − 210·35-s + 284·37-s + 8·41-s − 504·43-s − 280·47-s − 621·49-s + 150·53-s − 330·55-s − 350·59-s + 350·61-s + 120·65-s − 804·67-s − 500·71-s − 486·73-s + 308·77-s − 1.59e3·79-s + 684·83-s + ⋯ |
L(s) = 1 | + 1.34·5-s − 0.755·7-s − 0.603·11-s + 0.170·13-s + 0.485·17-s + 0.0482·19-s − 1.59·23-s + 6/5·25-s − 0.627·29-s − 0.509·31-s − 1.01·35-s + 1.26·37-s + 0.0304·41-s − 1.78·43-s − 0.868·47-s − 1.81·49-s + 0.388·53-s − 0.809·55-s − 0.772·59-s + 0.734·61-s + 0.228·65-s − 1.46·67-s − 0.835·71-s − 0.779·73-s + 0.455·77-s − 2.26·79-s + 0.904·83-s + ⋯ |
Λ(s)=(=((215⋅36⋅53)s/2ΓC(s)3L(s)−Λ(4−s)
Λ(s)=(=((215⋅36⋅53)s/2ΓC(s+3/2)3L(s)−Λ(1−s)
Degree: |
6 |
Conductor: |
215⋅36⋅53
|
Sign: |
−1
|
Analytic conductor: |
613317. |
Root analytic conductor: |
9.21752 |
Motivic weight: |
3 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
3
|
Selberg data: |
(6, 215⋅36⋅53, ( :3/2,3/2,3/2), −1)
|
Particular Values
L(2) |
= |
0 |
L(21) |
= |
0 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 3 | | 1 |
| 5 | C1 | (1−pT)3 |
good | 7 | S4×C2 | 1+2pT+817T2+9196T3+817p3T4+2p7T5+p9T6 |
| 11 | S4×C2 | 1+2pT+2149T2+69196T3+2149p3T4+2p7T5+p9T6 |
| 13 | S4×C2 | 1−8T+2127T2−119632T3+2127p3T4−8p6T5+p9T6 |
| 17 | S4×C2 | 1−2pT+7951T2−430972T3+7951p3T4−2p7T5+p9T6 |
| 19 | S4×C2 | 1−4T+7649T2−529240T3+7649p3T4−4p6T5+p9T6 |
| 23 | S4×C2 | 1+176T+35205T2+4252064T3+35205p3T4+176p6T5+p9T6 |
| 29 | S4×C2 | 1+98T+24635T2+7058028T3+24635p3T4+98p6T5+p9T6 |
| 31 | S4×C2 | 1+88T+52685T2+3535312T3+52685p3T4+88p6T5+p9T6 |
| 37 | S4×C2 | 1−284T+49559T2−13028696T3+49559p3T4−284p6T5+p9T6 |
| 41 | S4×C2 | 1−8T+121451T2−10434960T3+121451p3T4−8p6T5+p9T6 |
| 43 | S4×C2 | 1+504T+174009T2+40942288T3+174009p3T4+504p6T5+p9T6 |
| 47 | S4×C2 | 1+280T+115997T2+54406224T3+115997p3T4+280p6T5+p9T6 |
| 53 | S4×C2 | 1−150T+253683T2−44718980T3+253683p3T4−150p6T5+p9T6 |
| 59 | S4×C2 | 1+350T+202005T2−17161444T3+202005p3T4+350p6T5+p9T6 |
| 61 | S4×C2 | 1−350T+204635T2−134954132T3+204635p3T4−350p6T5+p9T6 |
| 67 | S4×C2 | 1+12pT+1077441T2+489158232T3+1077441p3T4+12p7T5+p9T6 |
| 71 | S4×C2 | 1+500T+1072245T2+353953688T3+1072245p3T4+500p6T5+p9T6 |
| 73 | S4×C2 | 1+486T+1084311T2+377044724T3+1084311p3T4+486p6T5+p9T6 |
| 79 | S4×C2 | 1+1592T+2161789T2+1645964560T3+2161789p3T4+1592p6T5+p9T6 |
| 83 | S4×C2 | 1−684T+1381713T2−561532168T3+1381713p3T4−684p6T5+p9T6 |
| 89 | S4×C2 | 1−668T+2135307T2−937072184T3+2135307p3T4−668p6T5+p9T6 |
| 97 | S4×C2 | 1+1394T+2868623T2+2439375164T3+2868623p3T4+1394p6T5+p9T6 |
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L(s)=p∏ j=1∏6(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−8.530035457335660671347475402042, −8.171484469528509396851308363737, −8.061573897460417842583104908360, −7.67539048431542725388536399193, −7.36443228676133439221025434157, −7.24586793390817502779695517051, −6.72863856426024891884689082320, −6.45483993093291754260281462720, −6.24006443379490506803945415717, −6.13163026196531398169423075138, −5.66504569079821394804361970927, −5.59232907393231261345645768659, −5.18932707492727457907114223289, −4.90377693494979912030258726224, −4.49524381585693470399921055117, −4.37432028201329440639032848434, −3.58521276633886909727825683369, −3.55490218119831981143907248424, −3.39496139058713302190351582471, −2.56918487757764805390665128746, −2.50291738472753237903198838970, −2.42806249527417820001900267067, −1.43616967379359230042493228342, −1.42677167007891290358703017084, −1.33579470341932987631431728676, 0, 0, 0,
1.33579470341932987631431728676, 1.42677167007891290358703017084, 1.43616967379359230042493228342, 2.42806249527417820001900267067, 2.50291738472753237903198838970, 2.56918487757764805390665128746, 3.39496139058713302190351582471, 3.55490218119831981143907248424, 3.58521276633886909727825683369, 4.37432028201329440639032848434, 4.49524381585693470399921055117, 4.90377693494979912030258726224, 5.18932707492727457907114223289, 5.59232907393231261345645768659, 5.66504569079821394804361970927, 6.13163026196531398169423075138, 6.24006443379490506803945415717, 6.45483993093291754260281462720, 6.72863856426024891884689082320, 7.24586793390817502779695517051, 7.36443228676133439221025434157, 7.67539048431542725388536399193, 8.061573897460417842583104908360, 8.171484469528509396851308363737, 8.530035457335660671347475402042