L(s) = 1 | + 2·3-s − 2·5-s + 20·7-s − 3·9-s + 22·11-s − 80·13-s − 4·15-s − 124·17-s − 72·19-s + 40·21-s − 98·23-s − 55·25-s + 34·27-s − 144·29-s − 34·31-s + 44·33-s − 40·35-s − 54·37-s − 160·39-s + 536·41-s + 60·43-s + 6·45-s − 272·47-s − 338·49-s − 248·51-s + 492·53-s − 44·55-s + ⋯ |
L(s) = 1 | + 0.384·3-s − 0.178·5-s + 1.07·7-s − 1/9·9-s + 0.603·11-s − 1.70·13-s − 0.0688·15-s − 1.76·17-s − 0.869·19-s + 0.415·21-s − 0.888·23-s − 0.439·25-s + 0.242·27-s − 0.922·29-s − 0.196·31-s + 0.232·33-s − 0.193·35-s − 0.239·37-s − 0.656·39-s + 2.04·41-s + 0.212·43-s + 0.0198·45-s − 0.844·47-s − 0.985·49-s − 0.680·51-s + 1.27·53-s − 0.107·55-s + ⋯ |
Λ(s)=(=(495616s/2ΓC(s)2L(s)Λ(4−s)
Λ(s)=(=(495616s/2ΓC(s+3/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
495616
= 212⋅112
|
Sign: |
1
|
Analytic conductor: |
1725.35 |
Root analytic conductor: |
6.44494 |
Motivic weight: |
3 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
2
|
Selberg data: |
(4, 495616, ( :3/2,3/2), 1)
|
Particular Values
L(2) |
= |
0 |
L(21) |
= |
0 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 11 | C1 | (1−pT)2 |
good | 3 | D4 | 1−2T+7T2−2p3T3+p6T4 |
| 5 | D4 | 1+2T+59T2+2p3T3+p6T4 |
| 7 | D4 | 1−20T+738T2−20p3T3+p6T4 |
| 13 | D4 | 1+80T+4794T2+80p3T3+p6T4 |
| 17 | D4 | 1+124T+13238T2+124p3T3+p6T4 |
| 19 | D4 | 1+72T+4214T2+72p3T3+p6T4 |
| 23 | D4 | 1+98T+22847T2+98p3T3+p6T4 |
| 29 | D4 | 1+144T+44554T2+144p3T3+p6T4 |
| 31 | D4 | 1+34T+57519T2+34p3T3+p6T4 |
| 37 | D4 | 1+54T+101843T2+54p3T3+p6T4 |
| 41 | D4 | 1−536T+209618T2−536p3T3+p6T4 |
| 43 | D4 | 1−60T+159146T2−60p3T3+p6T4 |
| 47 | D4 | 1+272T+182942T2+272p3T3+p6T4 |
| 53 | D4 | 1−492T+348862T2−492p3T3+p6T4 |
| 59 | D4 | 1+634T+458975T2+634p3T3+p6T4 |
| 61 | D4 | 1+840T+528794T2+840p3T3+p6T4 |
| 67 | D4 | 1+754T+742455T2+754p3T3+p6T4 |
| 71 | D4 | 1+678T+813415T2+678p3T3+p6T4 |
| 73 | D4 | 1+400T+160962T2+400p3T3+p6T4 |
| 79 | D4 | 1−4pT−279966T2−4p4T3+p6T4 |
| 83 | D4 | 1+468T+1155130T2+468p3T3+p6T4 |
| 89 | D4 | 1+1842T+1935427T2+1842p3T3+p6T4 |
| 97 | D4 | 1−2194T+2966547T2−2194p3T3+p6T4 |
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
−9.659890211653806100668861196911, −9.337001964357384850338755548042, −9.021088455503012753060679925998, −8.549173266722828068742926250328, −8.068786073483896527883817655673, −7.67320478042031699270768780324, −7.33516177957734884300535111967, −6.82673914574715326162010760868, −6.28038038251687987262360181879, −5.82693718417527506000725229542, −5.22613680538853857620486748250, −4.45217106719644271208490575739, −4.45122371502964288719910280408, −4.00134385091175629378275306062, −2.94621839644524322570989992619, −2.55201582993080766996807852719, −1.86280330592618412142484737768, −1.52242411151265508876516760398, 0, 0,
1.52242411151265508876516760398, 1.86280330592618412142484737768, 2.55201582993080766996807852719, 2.94621839644524322570989992619, 4.00134385091175629378275306062, 4.45122371502964288719910280408, 4.45217106719644271208490575739, 5.22613680538853857620486748250, 5.82693718417527506000725229542, 6.28038038251687987262360181879, 6.82673914574715326162010760868, 7.33516177957734884300535111967, 7.67320478042031699270768780324, 8.068786073483896527883817655673, 8.549173266722828068742926250328, 9.021088455503012753060679925998, 9.337001964357384850338755548042, 9.659890211653806100668861196911