L(s) = 1 | − 18·3-s − 6·5-s + 81·9-s + 90·11-s − 1.53e3·13-s + 108·15-s + 1.92e3·17-s + 2.24e3·19-s − 6.35e3·23-s + 690·25-s + 1.45e3·27-s + 2.11e4·29-s − 3.31e3·31-s − 1.62e3·33-s − 2.10e3·37-s + 2.76e4·39-s − 2.53e3·41-s − 1.15e4·43-s − 486·45-s + 1.56e4·47-s − 3.46e4·51-s − 1.65e4·53-s − 540·55-s − 4.04e4·57-s − 1.31e4·59-s − 5.79e3·61-s + 9.21e3·65-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.107·5-s + 1/3·9-s + 0.224·11-s − 2.52·13-s + 0.123·15-s + 1.61·17-s + 1.42·19-s − 2.50·23-s + 0.220·25-s + 0.384·27-s + 4.66·29-s − 0.618·31-s − 0.258·33-s − 0.252·37-s + 2.91·39-s − 0.235·41-s − 0.951·43-s − 0.0357·45-s + 1.03·47-s − 1.86·51-s − 0.807·53-s − 0.0240·55-s − 1.64·57-s − 0.491·59-s − 0.199·61-s + 0.270·65-s + ⋯ |
Λ(s)=(=((28⋅34⋅78)s/2ΓC(s)4L(s)Λ(6−s)
Λ(s)=(=((28⋅34⋅78)s/2ΓC(s+5/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
28⋅34⋅78
|
Sign: |
1
|
Analytic conductor: |
7.90954×107 |
Root analytic conductor: |
9.71111 |
Motivic weight: |
5 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(8, 28⋅34⋅78, ( :5/2,5/2,5/2,5/2), 1)
|
Particular Values
L(3) |
≈ |
0.5979498803 |
L(21) |
≈ |
0.5979498803 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 3 | C2 | (1+p2T+p4T2)2 |
| 7 | | 1 |
good | 5 | D4×C2 | 1+6T−654T2−6672pT3−376081p2T4−6672p6T5−654p10T6+6p15T7+p20T8 |
| 11 | D4×C2 | 1−90T−43146T2+24377040T3−24615785185T4+24377040p5T5−43146p10T6−90p15T7+p20T8 |
| 13 | D4 | (1+768T+689558T2+768p5T3+p10T4)2 |
| 17 | D4×C2 | 1−1926T−52038T2−1775386800T3+6866068206815T4−1775386800p5T5−52038p10T6−1926p15T7+p20T8 |
| 19 | D4×C2 | 1−2248T+2045674T2+4370939264T3−9597001149797T4+4370939264p5T5+2045674p10T6−2248p15T7+p20T8 |
| 23 | D4×C2 | 1+6354T+17412870T2+64097627040T3+225899059481879T4+64097627040p5T5+17412870p10T6+6354p15T7+p20T8 |
| 29 | D4 | (1−10572T+60053694T2−10572p5T3+p10T4)2 |
| 31 | D4×C2 | 1+3312T−20161598T2−86533816320T3−164535742920381T4−86533816320p5T5−20161598p10T6+3312p15T7+p20T8 |
| 37 | D4×C2 | 1+2104T−10065302T2−261307954784T3−4905535826642837T4−261307954784p5T5−10065302p10T6+2104p15T7+p20T8 |
| 41 | D4 | (1+1266T+226048450T2+1266p5T3+p10T4)2 |
| 43 | D4 | (1+5768T−18440058T2+5768p5T3+p10T4)2 |
| 47 | D4×C2 | 1−15612T−129736270T2+1330442150400T3+30982015974170739T4+1330442150400p5T5−129736270p10T6−15612p15T7+p20T8 |
| 53 | D4×C2 | 1+16512T−326970214T2−3909622657536T3+70632917811042123T4−3909622657536p5T5−326970214p10T6+16512p15T7+p20T8 |
| 59 | D4×C2 | 1+13140T−369558p2T2+384245136000T3+1494391243648374203T4+384245136000p5T5−369558p12T6+13140p15T7+p20T8 |
| 61 | D4×C2 | 1+5796T−1275860366T2−2200965041520T3+972953775501959307T4−2200965041520p5T5−1275860366p10T6+5796p15T7+p20T8 |
| 67 | D4×C2 | 1−56116T+850173514T2+22525987751552T3−789760835217853877T4+22525987751552p5T5+850173514p10T6−56116p15T7+p20T8 |
| 71 | D4 | (1−11022T−822078402T2−11022p5T3+p10T4)2 |
| 73 | D4×C2 | 1+85384T+1490284450T2+141225120630880T3+14230455000486300979T4+141225120630880p5T5+1490284450p10T6+85384p15T7+p20T8 |
| 79 | D4×C2 | 1−19620T−5831522702T2−1223391444480T3+27991691518342215603T4−1223391444480p5T5−5831522702p10T6−19620p15T7+p20T8 |
| 83 | D4 | (1−44424T+7801188630T2−44424p5T3+p10T4)2 |
| 89 | D4×C2 | 1−211218T+23178254114T2−2168505612203616T3+17⋯23T4−2168505612203616p5T5+23178254114p10T6−211218p15T7+p20T8 |
| 97 | D4 | (1+44864T+3170652414T2+44864p5T3+p10T4)2 |
show more | | |
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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
−6.88635258917440491606022157397, −6.56699996784460935574517593423, −6.39416791247236457151031355143, −6.21697453981815996563972831298, −5.89751611567900320596305456292, −5.78060334620223628473196694089, −5.21062515158322478743735948884, −5.19916182954355879991555913375, −5.06077940167807153813253860129, −4.71910719493133869426373524742, −4.64378193596545868062640878542, −4.29943828324176421824393290194, −3.82673394934010298392069978200, −3.64750509943567458252656660484, −3.33629306565436695452286483610, −3.02498987225598824498461885088, −2.62448035244952068694321690546, −2.49632413113200367686310444966, −2.33762802900758455706464986772, −1.59804112156089165665430964074, −1.55446426082211897692312589327, −1.02062251940713316132239054433, −0.799937453075725890137391300786, −0.52645699331340661634076803105, −0.11862014060307787771893686478,
0.11862014060307787771893686478, 0.52645699331340661634076803105, 0.799937453075725890137391300786, 1.02062251940713316132239054433, 1.55446426082211897692312589327, 1.59804112156089165665430964074, 2.33762802900758455706464986772, 2.49632413113200367686310444966, 2.62448035244952068694321690546, 3.02498987225598824498461885088, 3.33629306565436695452286483610, 3.64750509943567458252656660484, 3.82673394934010298392069978200, 4.29943828324176421824393290194, 4.64378193596545868062640878542, 4.71910719493133869426373524742, 5.06077940167807153813253860129, 5.19916182954355879991555913375, 5.21062515158322478743735948884, 5.78060334620223628473196694089, 5.89751611567900320596305456292, 6.21697453981815996563972831298, 6.39416791247236457151031355143, 6.56699996784460935574517593423, 6.88635258917440491606022157397