L(s) = 1 | − 4·2-s + 4-s + 14·7-s + 24·8-s + 92·11-s − 8·13-s − 56·14-s − 47·16-s − 44·17-s − 108·19-s − 368·22-s − 320·23-s + 32·26-s + 14·28-s + 236·29-s − 60·31-s + 52·32-s + 176·34-s − 204·37-s + 432·38-s − 44·41-s − 136·43-s + 92·44-s + 1.28e3·46-s + 400·47-s + 147·49-s − 8·52-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 1/8·4-s + 0.755·7-s + 1.06·8-s + 2.52·11-s − 0.170·13-s − 1.06·14-s − 0.734·16-s − 0.627·17-s − 1.30·19-s − 3.56·22-s − 2.90·23-s + 0.241·26-s + 0.0944·28-s + 1.51·29-s − 0.347·31-s + 0.287·32-s + 0.887·34-s − 0.906·37-s + 1.84·38-s − 0.167·41-s − 0.482·43-s + 0.315·44-s + 4.10·46-s + 1.24·47-s + 3/7·49-s − 0.0213·52-s + ⋯ |
Λ(s)=(=(2480625s/2ΓC(s)2L(s)Λ(4−s)
Λ(s)=(=(2480625s/2ΓC(s+3/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
2480625
= 34⋅54⋅72
|
Sign: |
1
|
Analytic conductor: |
8635.61 |
Root analytic conductor: |
9.63991 |
Motivic weight: |
3 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 2480625, ( :3/2,3/2), 1)
|
Particular Values
L(2) |
≈ |
1.050428496 |
L(21) |
≈ |
1.050428496 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 3 | | 1 |
| 5 | | 1 |
| 7 | C1 | (1−pT)2 |
good | 2 | D4 | 1+p2T+15T2+p5T3+p6T4 |
| 11 | D4 | 1−92T+4758T2−92p3T3+p6T4 |
| 13 | D4 | 1+8T−2810T2+8p3T3+p6T4 |
| 17 | D4 | 1+44T+630T2+44p3T3+p6T4 |
| 19 | D4 | 1+108T+13254T2+108p3T3+p6T4 |
| 23 | D4 | 1+320T+47934T2+320p3T3+p6T4 |
| 29 | D4 | 1−236T+61982T2−236p3T3+p6T4 |
| 31 | D4 | 1+60T+8462T2+60p3T3+p6T4 |
| 37 | D4 | 1+204T+108830T2+204p3T3+p6T4 |
| 41 | D4 | 1+44T+106326T2+44p3T3+p6T4 |
| 43 | D4 | 1+136T+81718T2+136p3T3+p6T4 |
| 47 | D4 | 1−400T+106526T2−400p3T3+p6T4 |
| 53 | D4 | 1−16T+260838T2−16p3T3+p6T4 |
| 59 | D4 | 1−464T+458102T2−464p3T3+p6T4 |
| 61 | D4 | 1+684T+535646T2+684p3T3+p6T4 |
| 67 | D4 | 1+736T+602470T2+736p3T3+p6T4 |
| 71 | D4 | 1−740T+757502T2−740p3T3+p6T4 |
| 73 | D4 | 1+424T+748558T2+424p3T3+p6T4 |
| 79 | D4 | 1+408T−143586T2+408p3T3+p6T4 |
| 83 | D4 | 1−608T+1200710T2−608p3T3+p6T4 |
| 89 | D4 | 1−1332T+1790774T2−1332p3T3+p6T4 |
| 97 | D4 | 1−2448T+3286542T2−2448p3T3+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
−9.031090576641441373614635692617, −8.845604869942392620376622277370, −8.514297371156935355032439595201, −8.474341455024723470936567120092, −7.68791290025835345030087649929, −7.54579351967256139237609065868, −6.95784187454072987476040509442, −6.34629257629547888247689509972, −6.20913155857813730264677779498, −5.91615389287345527139436811812, −4.79154641553107522513901292180, −4.76087308741234609122526767407, −4.13148405319743605170495231426, −3.92538522091243805615370411817, −3.40903490882162041523356912234, −2.34652686206042097081089699799, −1.85504130551169329715287371316, −1.61170418332528786560906348292, −0.815818025496236676430602633103, −0.39219303726823274276798959071,
0.39219303726823274276798959071, 0.815818025496236676430602633103, 1.61170418332528786560906348292, 1.85504130551169329715287371316, 2.34652686206042097081089699799, 3.40903490882162041523356912234, 3.92538522091243805615370411817, 4.13148405319743605170495231426, 4.76087308741234609122526767407, 4.79154641553107522513901292180, 5.91615389287345527139436811812, 6.20913155857813730264677779498, 6.34629257629547888247689509972, 6.95784187454072987476040509442, 7.54579351967256139237609065868, 7.68791290025835345030087649929, 8.474341455024723470936567120092, 8.514297371156935355032439595201, 8.845604869942392620376622277370, 9.031090576641441373614635692617