L(s) = 1 | + 4·5-s + 8·7-s − 92·11-s − 100·13-s − 92·17-s + 4·19-s + 8·23-s − 46·25-s + 84·29-s + 384·31-s + 32·35-s + 172·37-s + 300·41-s + 300·43-s − 16·47-s − 446·49-s − 12·53-s − 368·55-s + 644·59-s − 292·61-s − 400·65-s − 172·67-s + 408·71-s + 412·73-s − 736·77-s + 400·79-s − 948·83-s + ⋯ |
L(s) = 1 | + 0.357·5-s + 0.431·7-s − 2.52·11-s − 2.13·13-s − 1.31·17-s + 0.0482·19-s + 0.0725·23-s − 0.367·25-s + 0.537·29-s + 2.22·31-s + 0.154·35-s + 0.764·37-s + 1.14·41-s + 1.06·43-s − 0.0496·47-s − 1.30·49-s − 0.0311·53-s − 0.902·55-s + 1.42·59-s − 0.612·61-s − 0.763·65-s − 0.313·67-s + 0.681·71-s + 0.660·73-s − 1.08·77-s + 0.569·79-s − 1.25·83-s + ⋯ |
Λ(s)=(=(1327104s/2ΓC(s)2L(s)Λ(4−s)
Λ(s)=(=(1327104s/2ΓC(s+3/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
1327104
= 214⋅34
|
Sign: |
1
|
Analytic conductor: |
4619.94 |
Root analytic conductor: |
8.24440 |
Motivic weight: |
3 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 1327104, ( :3/2,3/2), 1)
|
Particular Values
L(2) |
≈ |
1.162221960 |
L(21) |
≈ |
1.162221960 |
L(25) |
|
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 | D4 | 1−4T+62T2−4p3T3+p6T4 |
| 7 | D4 | 1−8T+510T2−8p3T3+p6T4 |
| 11 | D4 | 1+92T+430pT2+92p3T3+p6T4 |
| 13 | D4 | 1+100T+5166T2+100p3T3+p6T4 |
| 17 | D4 | 1+92T+5030T2+92p3T3+p6T4 |
| 19 | D4 | 1−4T+11370T2−4p3T3+p6T4 |
| 23 | D4 | 1−8T+8798T2−8p3T3+p6T4 |
| 29 | D4 | 1−84T+45742T2−84p3T3+p6T4 |
| 31 | D4 | 1−384T+84158T2−384p3T3+p6T4 |
| 37 | D4 | 1−172T+99294T2−172p3T3+p6T4 |
| 41 | D4 | 1−300T+157270T2−300p3T3+p6T4 |
| 43 | D4 | 1−300T+180314T2−300p3T3+p6T4 |
| 47 | D4 | 1+16T+114782T2+16p3T3+p6T4 |
| 53 | D4 | 1+12T+288382T2+12p3T3+p6T4 |
| 59 | D4 | 1−644T+462170T2−644p3T3+p6T4 |
| 61 | D4 | 1+292T+240078T2+292p3T3+p6T4 |
| 67 | D4 | 1+172T+278250T2+172p3T3+p6T4 |
| 71 | D4 | 1−408T+672766T2−408p3T3+p6T4 |
| 73 | D4 | 1−412T+690678T2−412p3T3+p6T4 |
| 79 | D4 | 1−400T+963870T2−400p3T3+p6T4 |
| 83 | D4 | 1+948T+1360138T2+948p3T3+p6T4 |
| 89 | D4 | 1+572T+845846T2+572p3T3+p6T4 |
| 97 | D4 | 1−2204T+2633478T2−2204p3T3+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
−9.662792480496163355446543600710, −9.441459499161729544598997185917, −8.651910157887449987180556197540, −8.376341288831722219521912745059, −7.85479912013808595673996317229, −7.72141285147767170603437662495, −7.20166815985792174207873748113, −6.85663982471742886323434682254, −6.04895556470567799614040457354, −5.98136192580979019012121248397, −5.06944870754815578145752451427, −5.01924924228876027218602333715, −4.58891629868991337732050299903, −4.23099582514183572993488495344, −3.13682051332412651484221418709, −2.70864990701583260923350479167, −2.23537267017876954408340918840, −2.21298656889584361449449615550, −0.910587434348081480585361841018, −0.29964300976567435285467700142,
0.29964300976567435285467700142, 0.910587434348081480585361841018, 2.21298656889584361449449615550, 2.23537267017876954408340918840, 2.70864990701583260923350479167, 3.13682051332412651484221418709, 4.23099582514183572993488495344, 4.58891629868991337732050299903, 5.01924924228876027218602333715, 5.06944870754815578145752451427, 5.98136192580979019012121248397, 6.04895556470567799614040457354, 6.85663982471742886323434682254, 7.20166815985792174207873748113, 7.72141285147767170603437662495, 7.85479912013808595673996317229, 8.376341288831722219521912745059, 8.651910157887449987180556197540, 9.441459499161729544598997185917, 9.662792480496163355446543600710