L(s) = 1 | + 18·3-s − 78·5-s + 98·7-s + 243·9-s + 370·11-s − 1.05e3·13-s − 1.40e3·15-s − 1.02e3·17-s − 280·19-s + 1.76e3·21-s − 1.93e3·23-s + 3.13e3·25-s + 2.91e3·27-s + 2.55e3·29-s − 9.40e3·31-s + 6.66e3·33-s − 7.64e3·35-s − 168·37-s − 1.90e4·39-s − 8.16e3·41-s − 2.65e4·43-s − 1.89e4·45-s − 3.33e4·47-s + 7.20e3·49-s − 1.84e4·51-s + 3.59e4·53-s − 2.88e4·55-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 1.39·5-s + 0.755·7-s + 9-s + 0.921·11-s − 1.73·13-s − 1.61·15-s − 0.861·17-s − 0.177·19-s + 0.872·21-s − 0.760·23-s + 1.00·25-s + 0.769·27-s + 0.564·29-s − 1.75·31-s + 1.06·33-s − 1.05·35-s − 0.0201·37-s − 2.00·39-s − 0.758·41-s − 2.18·43-s − 1.39·45-s − 2.20·47-s + 3/7·49-s − 0.994·51-s + 1.75·53-s − 1.28·55-s + ⋯ |
Λ(s)=(=(112896s/2ΓC(s)2L(s)Λ(6−s)
Λ(s)=(=(112896s/2ΓC(s+5/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
112896
= 28⋅32⋅72
|
Sign: |
1
|
Analytic conductor: |
2904.02 |
Root analytic conductor: |
7.34091 |
Motivic weight: |
5 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
2
|
Selberg data: |
(4, 112896, ( :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 | | 1 |
| 3 | C1 | (1−p2T)2 |
| 7 | C1 | (1−p2T)2 |
good | 5 | D4 | 1+78T+2946T2+78p5T3+p10T4 |
| 11 | D4 | 1−370T+92110T2−370p5T3+p10T4 |
| 13 | D4 | 1+1056T+1002070T2+1056p5T3+p10T4 |
| 17 | D4 | 1+1026T+3101146T2+1026p5T3+p10T4 |
| 19 | D4 | 1+280T+4527126T2+280p5T3+p10T4 |
| 23 | D4 | 1+1930T+8676094T2+1930p5T3+p10T4 |
| 29 | D4 | 1−2556T+42210910T2−2556p5T3+p10T4 |
| 31 | D4 | 1+9408T+74445118T2+9408p5T3+p10T4 |
| 37 | D4 | 1+168T−25371242T2+168p5T3+p10T4 |
| 41 | D4 | 1+8166T+110576466T2+8166p5T3+p10T4 |
| 43 | D4 | 1+26552T+467848070T2+26552p5T3+p10T4 |
| 47 | D4 | 1+33324T+728437086T2+33324p5T3+p10T4 |
| 53 | D4 | 1−35904T+1106075878T2−35904p5T3+p10T4 |
| 59 | D4 | 1−332pT+633628534T2−332p6T3+p10T4 |
| 61 | D4 | 1+39028T+2066826686T2+39028p5T3+p10T4 |
| 67 | D4 | 1+57692T+3462397958T2+57692p5T3+p10T4 |
| 71 | D4 | 1+9558T+1530801150T2+9558p5T3+p10T4 |
| 73 | D4 | 1−8712T−1359024578T2−8712p5T3+p10T4 |
| 79 | D4 | 1+33420T+6193824670T2+33420p5T3+p10T4 |
| 83 | D4 | 1+17752T+7945747862T2+17752p5T3+p10T4 |
| 89 | D4 | 1+167718T+17824867954T2+167718p5T3+p10T4 |
| 97 | D4 | 1+46928T+163241630pT2+46928p5T3+p10T4 |
show more | | |
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L(s)=p∏ j=1∏4(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−10.31685558446154178310715702040, −10.05495717819009732396194948904, −9.483832325207236552175598675228, −8.924945292991519994465890686364, −8.519059911660735179091722664641, −8.244158889013104205174064241731, −7.62157680970586809830829996555, −7.31815335625968647044186042610, −6.87796205081820817524132961498, −6.36910601865660901372724013143, −5.22695060429923522237596925266, −4.89892099513590356478160263040, −4.18150844105286706236335997196, −3.98640605820107238500241488748, −3.24985450308796590422261284450, −2.65931988772819559974016718986, −1.85106190556346480537127101507, −1.45650547824126596360619039640, 0, 0,
1.45650547824126596360619039640, 1.85106190556346480537127101507, 2.65931988772819559974016718986, 3.24985450308796590422261284450, 3.98640605820107238500241488748, 4.18150844105286706236335997196, 4.89892099513590356478160263040, 5.22695060429923522237596925266, 6.36910601865660901372724013143, 6.87796205081820817524132961498, 7.31815335625968647044186042610, 7.62157680970586809830829996555, 8.244158889013104205174064241731, 8.519059911660735179091722664641, 8.924945292991519994465890686364, 9.483832325207236552175598675228, 10.05495717819009732396194948904, 10.31685558446154178310715702040