L(s) = 1 | + 8·2-s + 48·4-s − 108·5-s + 256·8-s − 864·10-s − 124·11-s + 720·13-s + 1.28e3·16-s + 1.26e3·17-s + 360·19-s − 5.18e3·20-s − 992·22-s − 6.52e3·23-s + 4.94e3·25-s + 5.76e3·26-s − 7.08e3·29-s + 5.90e3·31-s + 6.14e3·32-s + 1.00e4·34-s − 6.04e3·37-s + 2.88e3·38-s − 2.76e4·40-s + 1.73e4·41-s − 608·43-s − 5.95e3·44-s − 5.21e4·46-s + 3.04e4·47-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 3/2·4-s − 1.93·5-s + 1.41·8-s − 2.73·10-s − 0.308·11-s + 1.18·13-s + 5/4·16-s + 1.05·17-s + 0.228·19-s − 2.89·20-s − 0.436·22-s − 2.57·23-s + 1.58·25-s + 1.67·26-s − 1.56·29-s + 1.10·31-s + 1.06·32-s + 1.49·34-s − 0.725·37-s + 0.323·38-s − 2.73·40-s + 1.61·41-s − 0.0501·43-s − 0.463·44-s − 3.63·46-s + 2.01·47-s + ⋯ |
Λ(s)=(=(777924s/2ΓC(s)2L(s)Λ(6−s)
Λ(s)=(=(777924s/2ΓC(s+5/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
777924
= 22⋅34⋅74
|
Sign: |
1
|
Analytic conductor: |
20010.5 |
Root analytic conductor: |
11.8936 |
Motivic weight: |
5 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
2
|
Selberg data: |
(4, 777924, ( :5/2,5/2), 1)
|
Particular Values
L(3) |
= |
0 |
L(21) |
= |
0 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | C1 | (1−p2T)2 |
| 3 | | 1 |
| 7 | | 1 |
good | 5 | D4 | 1+108T+6716T2+108p5T3+p10T4 |
| 11 | D4 | 1+124T+294194T2+124p5T3+p10T4 |
| 13 | D4 | 1−720T+673736T2−720p5T3+p10T4 |
| 17 | D4 | 1−1260T+2796692T2−1260p5T3+p10T4 |
| 19 | D4 | 1−360T+4777230T2−360p5T3+p10T4 |
| 23 | D4 | 1+6524T+1009894pT2+6524p5T3+p10T4 |
| 29 | D4 | 1+7088T+48216146T2+7088p5T3+p10T4 |
| 31 | D4 | 1−5904T+58828406T2−5904p5T3+p10T4 |
| 37 | D4 | 1+6040T+111103002T2+6040p5T3+p10T4 |
| 41 | D4 | 1−17388T+216792980T2−17388p5T3+p10T4 |
| 43 | D4 | 1+608T+164053110T2+608p5T3+p10T4 |
| 47 | D4 | 1−648pT+629420198T2−648p6T3+p10T4 |
| 53 | D4 | 1+3964T+594431822T2+3964p5T3+p10T4 |
| 59 | D4 | 1+40752T+1841031182T2+40752p5T3+p10T4 |
| 61 | D4 | 1+1368T+1545466296T2+1368p5T3+p10T4 |
| 67 | D4 | 1+16224T+813179750T2+16224p5T3+p10T4 |
| 71 | D4 | 1−3204T−212201462T2−3204p5T3+p10T4 |
| 73 | D4 | 1−23976T+68704368T2−23976p5T3+p10T4 |
| 79 | D4 | 1+1040pT+7785668670T2+1040p6T3+p10T4 |
| 83 | D4 | 1+173736T+14732805782T2+173736p5T3+p10T4 |
| 89 | D4 | 1+200556T+21158382260T2+200556p5T3+p10T4 |
| 97 | D4 | 1−251928T+32348722272T2−251928p5T3+p10T4 |
show more | | |
show less | | |
L(s)=p∏ j=1∏4(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−9.047409172072625776713966181973, −8.590615522744635089646514918710, −8.016107556012015818949064595035, −7.88277245126493696555168659416, −7.38961933418289585681318955410, −7.28099661275107127614004906919, −6.31815670830155299166156541536, −6.14966986664153974324857214883, −5.54716310709169844178730456079, −5.36774037980630252284283547740, −4.37672752137482228456574187899, −4.20775747082601207721577958367, −3.74490234916446448722665794607, −3.68192868222775962197203958632, −2.81355463206960669057603142382, −2.51705136803645793995081085002, −1.47499430740729147525165745607, −1.19102157271984553626278081789, 0, 0,
1.19102157271984553626278081789, 1.47499430740729147525165745607, 2.51705136803645793995081085002, 2.81355463206960669057603142382, 3.68192868222775962197203958632, 3.74490234916446448722665794607, 4.20775747082601207721577958367, 4.37672752137482228456574187899, 5.36774037980630252284283547740, 5.54716310709169844178730456079, 6.14966986664153974324857214883, 6.31815670830155299166156541536, 7.28099661275107127614004906919, 7.38961933418289585681318955410, 7.88277245126493696555168659416, 8.016107556012015818949064595035, 8.590615522744635089646514918710, 9.047409172072625776713966181973