L(s) = 1 | − 20·5-s − 14·7-s − 4·11-s + 30·13-s + 28·17-s − 78·19-s + 182·23-s + 250·25-s − 202·29-s + 76·31-s + 280·35-s + 302·37-s − 380·41-s − 178·43-s + 114·47-s − 109·49-s + 256·53-s + 80·55-s − 204·59-s + 766·61-s − 600·65-s − 330·67-s − 1.06e3·71-s + 1.44e3·73-s + 56·77-s − 742·79-s − 768·83-s + ⋯ |
L(s) = 1 | − 1.78·5-s − 0.755·7-s − 0.109·11-s + 0.640·13-s + 0.399·17-s − 0.941·19-s + 1.64·23-s + 2·25-s − 1.29·29-s + 0.440·31-s + 1.35·35-s + 1.34·37-s − 1.44·41-s − 0.631·43-s + 0.353·47-s − 0.317·49-s + 0.663·53-s + 0.196·55-s − 0.450·59-s + 1.60·61-s − 1.14·65-s − 0.601·67-s − 1.77·71-s + 2.31·73-s + 0.0828·77-s − 1.05·79-s − 1.01·83-s + ⋯ |
Λ(s)=(=((216⋅312⋅54)s/2ΓC(s)4L(s)Λ(4−s)
Λ(s)=(=((216⋅312⋅54)s/2ΓC(s+3/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
216⋅312⋅54
|
Sign: |
1
|
Analytic conductor: |
2.63802×108 |
Root analytic conductor: |
11.2891 |
Motivic weight: |
3 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
4
|
Selberg data: |
(8, 216⋅312⋅54, ( :3/2,3/2,3/2,3/2), 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)4 |
good | 7 | C2≀S4 | 1+2pT+305T2+4614T3+488pT4+4614p3T5+305p6T6+2p10T7+p12T8 |
| 11 | C2≀S4 | 1+4T+1575T2−29784T3+2457124T4−29784p3T5+1575p6T6+4p9T7+p12T8 |
| 13 | C2≀S4 | 1−30T−140T2−26820T3+7961973T4−26820p3T5−140p6T6−30p9T7+p12T8 |
| 17 | C2≀S4 | 1−28T+6099T2−159456T3+948020pT4−159456p3T5+6099p6T6−28p9T7+p12T8 |
| 19 | C2≀S4 | 1+78T+16433T2+1084170T3+152743716T4+1084170p3T5+16433p6T6+78p9T7+p12T8 |
| 23 | C2≀S4 | 1−182T+33285T2−2652138T3+367341256T4−2652138p3T5+33285p6T6−182p9T7+p12T8 |
| 29 | C2≀S4 | 1+202T+54117T2+3130494T3+888383644T4+3130494p3T5+54117p6T6+202p9T7+p12T8 |
| 31 | C2≀S4 | 1−76T+52175T2−3793644T3+2235420704T4−3793644p3T5+52175p6T6−76p9T7+p12T8 |
| 37 | C2≀S4 | 1−302T+144869T2−1237962pT3+9554438060T4−1237962p4T5+144869p6T6−302p9T7+p12T8 |
| 41 | C2≀S4 | 1+380T+304944T2+77291940T3+32535322366T4+77291940p3T5+304944p6T6+380p9T7+p12T8 |
| 43 | C2≀S4 | 1+178T+211805T2+53964162T3+20664465836T4+53964162p3T5+211805p6T6+178p9T7+p12T8 |
| 47 | C2≀S4 | 1−114T+308433T2−39725310T3+42951666476T4−39725310p3T5+308433p6T6−114p9T7+p12T8 |
| 53 | C2≀S4 | 1−256T+177264T2−81601152T3+34810114894T4−81601152p3T5+177264p6T6−256p9T7+p12T8 |
| 59 | C2≀S4 | 1+204T+83084T2−95352084T3−46391911626T4−95352084p3T5+83084p6T6+204p9T7+p12T8 |
| 61 | C2≀S4 | 1−766T+444473T2−98451618T3+48468291764T4−98451618p3T5+444473p6T6−766p9T7+p12T8 |
| 67 | C2≀S4 | 1+330T+1123349T2+294102570T3+495222287532T4+294102570p3T5+1123349p6T6+330p9T7+p12T8 |
| 71 | C2≀S4 | 1+1060T+1430856T2+1119382500T3+767140525006T4+1119382500p3T5+1430856p6T6+1060p9T7+p12T8 |
| 73 | C2≀S4 | 1−1442T+1995917T2−1634684910T3+1251343303316T4−1634684910p3T5+1995917p6T6−1442p9T7+p12T8 |
| 79 | C2≀S4 | 1+742T+759772T2+600002584T3+627086485909T4+600002584p3T5+759772p6T6+742p9T7+p12T8 |
| 83 | C2≀S4 | 1+768T+2075540T2+1295690112T3+1724588712726T4+1295690112p3T5+2075540p6T6+768p9T7+p12T8 |
| 89 | C2≀S4 | 1−400T+2456228T2−740686960T3+2486053553638T4−740686960p3T5+2456228p6T6−400p9T7+p12T8 |
| 97 | C2≀S4 | 1−3338T+6353893T2−8727311702T3+9281975879860T4−8727311702p3T5+6353893p6T6−3338p9T7+p12T8 |
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L(s)=p∏ j=1∏8(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−6.67013207047215842814041817779, −6.28437737909305034887817509677, −6.02754366729099058801556211595, −6.02416231020477782493989657313, −5.73537117788599248122141209537, −5.32430872511825248893531437693, −5.15748677329458019868069892075, −4.96926929817038081508935351691, −4.89345235458928274085556640474, −4.56304193048878833325781966818, −4.34088185540242154088978749853, −3.98172740002470697049815080978, −3.95703409423345648137826620107, −3.66188762588000779099928579359, −3.37022808491706242054982126823, −3.33854041190009849974516095342, −3.28231741178723094787648609750, −2.64057127089699552993416121136, −2.45109460755093593343389066785, −2.29964731430367246191271985723, −2.25668718417237057877709237878, −1.29226625441494257434469020694, −1.25009991303512396055323738024, −1.13885379776093629722219097671, −1.00013694736494124864172200017, 0, 0, 0, 0,
1.00013694736494124864172200017, 1.13885379776093629722219097671, 1.25009991303512396055323738024, 1.29226625441494257434469020694, 2.25668718417237057877709237878, 2.29964731430367246191271985723, 2.45109460755093593343389066785, 2.64057127089699552993416121136, 3.28231741178723094787648609750, 3.33854041190009849974516095342, 3.37022808491706242054982126823, 3.66188762588000779099928579359, 3.95703409423345648137826620107, 3.98172740002470697049815080978, 4.34088185540242154088978749853, 4.56304193048878833325781966818, 4.89345235458928274085556640474, 4.96926929817038081508935351691, 5.15748677329458019868069892075, 5.32430872511825248893531437693, 5.73537117788599248122141209537, 6.02416231020477782493989657313, 6.02754366729099058801556211595, 6.28437737909305034887817509677, 6.67013207047215842814041817779