L(s) = 1 | − 9·9-s − 72·11-s + 56·19-s + 372·29-s + 352·31-s − 60·41-s − 49·49-s + 1.12e3·59-s − 644·61-s − 96·71-s + 992·79-s + 81·81-s − 2.62e3·89-s + 648·99-s + 2.07e3·101-s + 1.07e3·109-s + 1.22e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.23e3·169-s + ⋯ |
L(s) = 1 | − 1/3·9-s − 1.97·11-s + 0.676·19-s + 2.38·29-s + 2.03·31-s − 0.228·41-s − 1/7·49-s + 2.48·59-s − 1.35·61-s − 0.160·71-s + 1.41·79-s + 1/9·81-s − 3.12·89-s + 0.657·99-s + 2.04·101-s + 0.945·109-s + 0.921·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.47·169-s + ⋯ |
Λ(s)=(=(4410000s/2ΓC(s)2L(s)Λ(4−s)
Λ(s)=(=(4410000s/2ΓC(s+3/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
4410000
= 24⋅32⋅54⋅72
|
Sign: |
1
|
Analytic conductor: |
15352.2 |
Root analytic conductor: |
11.1312 |
Motivic weight: |
3 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 4410000, ( :3/2,3/2), 1)
|
Particular Values
L(2) |
≈ |
3.239348776 |
L(21) |
≈ |
3.239348776 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 3 | C2 | 1+p2T2 |
| 5 | | 1 |
| 7 | C2 | 1+p2T2 |
good | 11 | C2 | (1+36T+p3T2)2 |
| 13 | C22 | 1−3238T2+p6T4 |
| 17 | C22 | 1−9790T2+p6T4 |
| 19 | C2 | (1−28T+p3T2)2 |
| 23 | C22 | 1+12530T2+p6T4 |
| 29 | C2 | (1−186T+p3T2)2 |
| 31 | C2 | (1−176T+p3T2)2 |
| 37 | C22 | 1+73418T2+p6T4 |
| 41 | C2 | (1+30T+p3T2)2 |
| 43 | C22 | 1+10730T2+p6T4 |
| 47 | C22 | 1−21022T2+p6T4 |
| 53 | C22 | 1−204118T2+p6T4 |
| 59 | C2 | (1−564T+p3T2)2 |
| 61 | C2 | (1+322T+p3T2)2 |
| 67 | C22 | 1−88870T2+p6T4 |
| 71 | C2 | (1+48T+p3T2)2 |
| 73 | C22 | 1+384050T2+p6T4 |
| 79 | C2 | (1−496T+p3T2)2 |
| 83 | C22 | 1−924550T2+p6T4 |
| 89 | C2 | (1+1314T+p3T2)2 |
| 97 | C22 | 1+242498T2+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
−8.734929431542620152009032501132, −8.386176837901720016061457975226, −8.272576904745783059630702332566, −7.935202350023116594696980665213, −7.33748571211491957759774723135, −7.13587130067315372046697426745, −6.47326311175567967049811749956, −6.31150237624100314912831340859, −5.64611996167613717596548255451, −5.42459525683894264807727473101, −4.78683324838058955662225251178, −4.74639242893538630970878805097, −4.16513354780198424437595825245, −3.43150032102024509899743071341, −2.90715903213987040638554415528, −2.74988301121841387995468060550, −2.28627856871080200591304076020, −1.52297557768678847880597076959, −0.70648461417939153523242630146, −0.53662744540970770279638032834,
0.53662744540970770279638032834, 0.70648461417939153523242630146, 1.52297557768678847880597076959, 2.28627856871080200591304076020, 2.74988301121841387995468060550, 2.90715903213987040638554415528, 3.43150032102024509899743071341, 4.16513354780198424437595825245, 4.74639242893538630970878805097, 4.78683324838058955662225251178, 5.42459525683894264807727473101, 5.64611996167613717596548255451, 6.31150237624100314912831340859, 6.47326311175567967049811749956, 7.13587130067315372046697426745, 7.33748571211491957759774723135, 7.935202350023116594696980665213, 8.272576904745783059630702332566, 8.386176837901720016061457975226, 8.734929431542620152009032501132