L(s) = 1 | − 92·5-s − 84·13-s − 1.92e3·17-s + 98·25-s + 5.10e3·29-s + 2.39e4·37-s + 1.01e4·41-s − 5.96e3·49-s + 3.94e4·53-s + 5.86e4·61-s + 7.72e3·65-s + 7.58e4·73-s + 1.77e5·85-s − 2.78e4·89-s + 3.27e5·97-s + 2.97e5·101-s + 2.46e5·109-s + 1.02e5·113-s − 3.15e5·121-s + 4.73e5·125-s + 127-s + 131-s + 137-s + 139-s − 4.69e5·145-s + 149-s + 151-s + ⋯ |
L(s) = 1 | − 1.64·5-s − 0.137·13-s − 1.61·17-s + 0.0313·25-s + 1.12·29-s + 2.87·37-s + 0.943·41-s − 0.354·49-s + 1.92·53-s + 2.01·61-s + 0.226·65-s + 1.66·73-s + 2.65·85-s − 0.372·89-s + 3.53·97-s + 2.89·101-s + 1.98·109-s + 0.755·113-s − 1.95·121-s + 2.70·125-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s − 1.85·145-s + 3.69e−6·149-s + 3.56e−6·151-s + ⋯ |
Λ(s)=(=(82944s/2ΓC(s)2L(s)Λ(6−s)
Λ(s)=(=(82944s/2ΓC(s+5/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
82944
= 210⋅34
|
Sign: |
1
|
Analytic conductor: |
2133.56 |
Root analytic conductor: |
6.79636 |
Motivic weight: |
5 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 82944, ( :5/2,5/2), 1)
|
Particular Values
L(3) |
≈ |
1.759320850 |
L(21) |
≈ |
1.759320850 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 3 | | 1 |
good | 5 | C2 | (1+46T+p5T2)2 |
| 7 | C22 | 1+5966T2+p10T4 |
| 11 | C22 | 1+315190T2+p10T4 |
| 13 | C2 | (1+42T+p5T2)2 |
| 17 | C2 | (1+962T+p5T2)2 |
| 19 | C22 | 1+632198T2+p10T4 |
| 23 | C22 | 1+2891758T2+p10T4 |
| 29 | C2 | (1−2554T+p5T2)2 |
| 31 | C22 | 1+53276990T2+p10T4 |
| 37 | C2 | (1−11950T+p5T2)2 |
| 41 | C2 | (1−5078T+p5T2)2 |
| 43 | C22 | 1+136416374T2+p10T4 |
| 47 | C22 | 1+307289566T2+p10T4 |
| 53 | C2 | (1−19714T+p5T2)2 |
| 59 | C22 | 1+1350713110T2+p10T4 |
| 61 | C2 | (1−29318T+p5T2)2 |
| 67 | C22 | 1+2415413606T2+p10T4 |
| 71 | C22 | 1−2948789810T2+p10T4 |
| 73 | C2 | (1−37914T+p5T2)2 |
| 79 | C22 | 1−1729880290T2+p10T4 |
| 83 | C22 | 1+6331666438T2+p10T4 |
| 89 | C2 | (1+13930T+p5T2)2 |
| 97 | C2 | (1−163602T+p5T2)2 |
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
−11.34383389752881068094036698900, −10.99049925932865542019027855420, −10.08404055037443828895290050110, −10.03949827383007200008594110614, −9.044993264307684688384853875608, −8.920107024939429195089905065997, −8.120770306359113842152721198164, −7.948374890925111551907800457713, −7.37579898530281104887808030468, −6.94464611028993456769383353896, −6.22569251421134925622080811415, −5.91110597448325712623150697382, −4.70845915791628599869013445088, −4.69104981662115926391125119210, −3.85673858882470782456961428808, −3.64722337738717004396749739984, −2.51079726718942078248683489188, −2.24122681438354862994139369915, −0.818292125960851381695758030822, −0.51034730826459727123039658323,
0.51034730826459727123039658323, 0.818292125960851381695758030822, 2.24122681438354862994139369915, 2.51079726718942078248683489188, 3.64722337738717004396749739984, 3.85673858882470782456961428808, 4.69104981662115926391125119210, 4.70845915791628599869013445088, 5.91110597448325712623150697382, 6.22569251421134925622080811415, 6.94464611028993456769383353896, 7.37579898530281104887808030468, 7.948374890925111551907800457713, 8.120770306359113842152721198164, 8.920107024939429195089905065997, 9.044993264307684688384853875608, 10.03949827383007200008594110614, 10.08404055037443828895290050110, 10.99049925932865542019027855420, 11.34383389752881068094036698900