L(s) = 1 | − 144·11-s − 104·19-s − 156·29-s + 240·31-s − 724·41-s + 670·49-s + 1.39e3·59-s + 444·61-s − 192·71-s + 1.26e3·79-s + 1.98e3·89-s − 1.78e3·101-s − 892·109-s + 1.28e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.35e3·169-s + 173-s + 179-s + 181-s + ⋯ |
L(s) = 1 | − 3.94·11-s − 1.25·19-s − 0.998·29-s + 1.39·31-s − 2.75·41-s + 1.95·49-s + 3.07·59-s + 0.931·61-s − 0.320·71-s + 1.80·79-s + 2.36·89-s − 1.75·101-s − 0.783·109-s + 9.68·121-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 + 1.98·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + ⋯ |
Λ(s)=(=(3240000s/2ΓC(s)2L(s)Λ(4−s)
Λ(s)=(=(3240000s/2ΓC(s+3/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
3240000
= 26⋅34⋅54
|
Sign: |
1
|
Analytic conductor: |
11279.1 |
Root analytic conductor: |
10.3055 |
Motivic weight: |
3 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 3240000, ( :3/2,3/2), 1)
|
Particular Values
L(2) |
≈ |
1.504305627 |
L(21) |
≈ |
1.504305627 |
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 |
| 5 | | 1 |
good | 7 | C22 | 1−670T2+p6T4 |
| 11 | C2 | (1+72T+p3T2)2 |
| 13 | C22 | 1−4358T2+p6T4 |
| 17 | C22 | 1−8382T2+p6T4 |
| 19 | C2 | (1+52T+p3T2)2 |
| 23 | C22 | 1−1230T2+p6T4 |
| 29 | C2 | (1+78T+p3T2)2 |
| 31 | C2 | (1−120T+p3T2)2 |
| 37 | C22 | 1−78806T2+p6T4 |
| 41 | C2 | (1+362T+p3T2)2 |
| 43 | C22 | 1+75242T2+p6T4 |
| 47 | C22 | 1−129246T2+p6T4 |
| 53 | C22 | 1+151146T2+p6T4 |
| 59 | C2 | (1−696T+p3T2)2 |
| 61 | C2 | (1−222T+p3T2)2 |
| 67 | C22 | 1−601510T2+p6T4 |
| 71 | C2 | (1+96T+p3T2)2 |
| 73 | C22 | 1−746350T2+p6T4 |
| 79 | C2 | (1−8pT+p3T2)2 |
| 83 | C22 | 1−769030T2+p6T4 |
| 89 | C2 | (1−994T+p3T2)2 |
| 97 | C22 | 1+844610T2+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
−8.769624283951839779224161818360, −8.729506132722121740935377533976, −8.252461804679939124160849011687, −8.021432699691598130693618774215, −7.47366694378677567527593638023, −7.41846790654937294001570853889, −6.61374709425533344248739659561, −6.47945172842430259768838614561, −5.67865959901195474963839717366, −5.38820423823188472610113684890, −5.04527079815134221666517897767, −4.93764322837246195963958473084, −4.04455543653042023952337869989, −3.77153685162631980518425940141, −2.86872127922451614923331089956, −2.77872204378069124863887952772, −2.12657878612317860353125078668, −1.95035263958724987888613994055, −0.60467369287496069402479059828, −0.40935333957269288356934686551,
0.40935333957269288356934686551, 0.60467369287496069402479059828, 1.95035263958724987888613994055, 2.12657878612317860353125078668, 2.77872204378069124863887952772, 2.86872127922451614923331089956, 3.77153685162631980518425940141, 4.04455543653042023952337869989, 4.93764322837246195963958473084, 5.04527079815134221666517897767, 5.38820423823188472610113684890, 5.67865959901195474963839717366, 6.47945172842430259768838614561, 6.61374709425533344248739659561, 7.41846790654937294001570853889, 7.47366694378677567527593638023, 8.021432699691598130693618774215, 8.252461804679939124160849011687, 8.729506132722121740935377533976, 8.769624283951839779224161818360