L(s) = 1 | − 33·2-s − 163·3-s + 235·4-s + 471·5-s + 5.37e3·6-s − 1.12e4·7-s − 5.32e3·8-s − 4.10e4·9-s − 1.55e4·10-s − 4.01e4·11-s − 3.83e4·12-s − 1.14e5·13-s + 3.70e5·14-s − 7.67e4·15-s + 2.42e5·16-s + 7.87e4·17-s + 1.35e6·18-s + 2.09e5·19-s + 1.10e5·20-s + 1.83e6·21-s + 1.32e6·22-s − 4.25e6·23-s + 8.67e5·24-s − 5.24e6·25-s + 3.77e6·26-s + 8.51e6·27-s − 2.64e6·28-s + ⋯ |
L(s) = 1 | − 1.45·2-s − 1.16·3-s + 0.458·4-s + 0.337·5-s + 1.69·6-s − 1.76·7-s − 0.459·8-s − 2.08·9-s − 0.491·10-s − 0.826·11-s − 0.533·12-s − 1.10·13-s + 2.58·14-s − 0.391·15-s + 0.925·16-s + 0.228·17-s + 3.04·18-s + 0.369·19-s + 0.154·20-s + 2.05·21-s + 1.20·22-s − 3.17·23-s + 0.534·24-s − 2.68·25-s + 1.61·26-s + 3.08·27-s − 0.812·28-s + ⋯ |
Λ(s)=(=(28561s/2ΓC(s)4L(s)Λ(10−s)
Λ(s)=(=(28561s/2ΓC(s+9/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
28561
= 134
|
Sign: |
1
|
Analytic conductor: |
2009.66 |
Root analytic conductor: |
2.58755 |
Motivic weight: |
9 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
4
|
Selberg data: |
(8, 28561, ( :9/2,9/2,9/2,9/2), 1)
|
Particular Values
L(5) |
= |
0 |
L(21) |
= |
0 |
L(211) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 13 | C1 | (1+p4T)4 |
good | 2 | C2≀S4 | 1+33T+427pT2+3219p3T3+9097p6T4+3219p12T5+427p19T6+33p27T7+p36T8 |
| 3 | C2≀S4 | 1+163T+67627T2+3067592pT3+212179120p2T4+3067592p10T5+67627p18T6+163p27T7+p36T8 |
| 5 | C2≀S4 | 1−471T+5467249T2−2978826294T3+2762975487438pT4−2978826294p9T5+5467249p18T6−471p27T7+p36T8 |
| 7 | C2≀S4 | 1+11241T+18891399pT2+2792105460p3T3+22711650939506p3T4+2792105460p12T5+18891399p19T6+11241p27T7+p36T8 |
| 11 | C2≀S4 | 1+40140T+5207212188T2+260939722519836T3+16186230268309124390T4+260939722519836p9T5+5207212188p18T6+40140p27T7+p36T8 |
| 17 | C2≀S4 | 1−78717T+179543541353T2+34209748146126354T3+16⋯82T4+34209748146126354p9T5+179543541353p18T6−78717p27T7+p36T8 |
| 19 | C2≀S4 | 1−209664T+58556387148pT2−9405441041308416pT3+51⋯22T4−9405441041308416p10T5+58556387148p19T6−209664p27T7+p36T8 |
| 23 | C2≀S4 | 1+4257444T+7366209360108T2+5287090655000044788T3+22⋯34T4+5287090655000044788p9T5+7366209360108p18T6+4257444p27T7+p36T8 |
| 29 | C2≀S4 | 1+1647936T+37315906740828T2+63599271454922849280T3+66⋯74T4+63599271454922849280p9T5+37315906740828p18T6+1647936p27T7+p36T8 |
| 31 | C2≀S4 | 1+11366002T+118630502542644T2+68⋯18T3+42⋯06T4+68⋯18p9T5+118630502542644p18T6+11366002p27T7+p36T8 |
| 37 | C2≀S4 | 1−4636891T+361056879801945T2−14⋯94T3+66⋯34T4−14⋯94p9T5+361056879801945p18T6−4636891p27T7+p36T8 |
| 41 | C2≀S4 | 1−13859538T+1246904510238356T2−13⋯86T3+60⋯42T4−13⋯86p9T5+1246904510238356p18T6−13859538p27T7+p36T8 |
| 43 | C2≀S4 | 1+33368081T+1484927456162487T2+42⋯12T3+10⋯24T4+42⋯12p9T5+1484927456162487p18T6+33368081p27T7+p36T8 |
| 47 | C2≀S4 | 1+3943005T+2343750668926273T2+81⋯96T3+37⋯54T4+81⋯96p9T5+2343750668926273p18T6+3943005p27T7+p36T8 |
| 53 | C2≀S4 | 1+171019326T+20398534023554412T2+16⋯86T3+11⋯66T4+16⋯86p9T5+20398534023554412p18T6+171019326p27T7+p36T8 |
| 59 | C2≀S4 | 1+63389388T+18604706307570428T2+24⋯68T3+14⋯74T4+24⋯68p9T5+18604706307570428p18T6+63389388p27T7+p36T8 |
| 61 | C2≀S4 | 1−77050190T+39401474083082580T2−26⋯86T3+65⋯70T4−26⋯86p9T5+39401474083082580p18T6−77050190p27T7+p36T8 |
| 67 | C2≀S4 | 1+41174072T+67202484458082900T2+19⋯28T3+23⋯46T4+19⋯28p9T5+67202484458082900p18T6+41174072p27T7+p36T8 |
| 71 | C2≀S4 | 1−252460989T+144403365822266409T2−44⋯68pT3+90⋯90T4−44⋯68p10T5+144403365822266409p18T6−252460989p27T7+p36T8 |
| 73 | C2≀S4 | 1−594415068T+287796385433349156T2−92⋯76T3+26⋯46T4−92⋯76p9T5+287796385433349156p18T6−594415068p27T7+p36T8 |
| 79 | C2≀S4 | 1−115998984T+371265024783771132T2−33⋯80T3+62⋯58T4−33⋯80p9T5+371265024783771132p18T6−115998984p27T7+p36T8 |
| 83 | C2≀S4 | 1+79577862T+6204891761421028pT2+86⋯46T3+12⋯38T4+86⋯46p9T5+6204891761421028p19T6+79577862p27T7+p36T8 |
| 89 | C2≀S4 | 1+1152240276T+746897787143209636T2+27⋯00T3−70⋯14T4+27⋯00p9T5+746897787143209636p18T6+1152240276p27T7+p36T8 |
| 97 | C2≀S4 | 1−1049098084T+2374666548574190388T2−23⋯60T3+24⋯74T4−23⋯60p9T5+2374666548574190388p18T6−1049098084p27T7+p36T8 |
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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
−13.95928460430890317870092855744, −12.97106901514597117453914352925, −12.85445491453793972570809894225, −12.43626443260267289556638154263, −11.83918470565123820424209471048, −11.75124371675164354200421837086, −11.31969655516604555701088514367, −10.94595480247553116440448571098, −10.26495302653789196245989292064, −9.892948414715274431333698709278, −9.608986960805973076664105819727, −9.274698042266345077491294788431, −9.196526681358649258536784772497, −8.044205686931871745573286906803, −8.039987945857310034350189533328, −7.82905030775083749862309007042, −6.57639634467340707158950518251, −6.31081068795648558770452480345, −5.84923773055124293501359703709, −5.54522832179349602028947311014, −5.19631160721890757832532363900, −3.73592644965051715793501526344, −3.38532871414856603989847282361, −2.51114059648194216727569569543, −1.97447730592761554791926188116, 0, 0, 0, 0,
1.97447730592761554791926188116, 2.51114059648194216727569569543, 3.38532871414856603989847282361, 3.73592644965051715793501526344, 5.19631160721890757832532363900, 5.54522832179349602028947311014, 5.84923773055124293501359703709, 6.31081068795648558770452480345, 6.57639634467340707158950518251, 7.82905030775083749862309007042, 8.039987945857310034350189533328, 8.044205686931871745573286906803, 9.196526681358649258536784772497, 9.274698042266345077491294788431, 9.608986960805973076664105819727, 9.892948414715274431333698709278, 10.26495302653789196245989292064, 10.94595480247553116440448571098, 11.31969655516604555701088514367, 11.75124371675164354200421837086, 11.83918470565123820424209471048, 12.43626443260267289556638154263, 12.85445491453793972570809894225, 12.97106901514597117453914352925, 13.95928460430890317870092855744