L(s) = 1 | + 8·2-s − 3·3-s + 48·4-s − 24·6-s − 114·7-s + 256·8-s − 119·9-s + 661·11-s − 144·12-s − 1.61e3·13-s − 912·14-s + 1.28e3·16-s − 64·17-s − 952·18-s + 722·19-s + 342·21-s + 5.28e3·22-s + 3.18e3·23-s − 768·24-s − 1.29e4·26-s + 12·27-s − 5.47e3·28-s − 2.48e3·29-s − 1.18e3·31-s + 6.14e3·32-s − 1.98e3·33-s − 512·34-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 0.192·3-s + 3/2·4-s − 0.272·6-s − 0.879·7-s + 1.41·8-s − 0.489·9-s + 1.64·11-s − 0.288·12-s − 2.64·13-s − 1.24·14-s + 5/4·16-s − 0.0537·17-s − 0.692·18-s + 0.458·19-s + 0.169·21-s + 2.32·22-s + 1.25·23-s − 0.272·24-s − 3.74·26-s + 0.00316·27-s − 1.31·28-s − 0.547·29-s − 0.220·31-s + 1.06·32-s − 0.316·33-s − 0.0759·34-s + ⋯ |
Λ(s)=(=(902500s/2ΓC(s)2L(s)Λ(6−s)
Λ(s)=(=(902500s/2ΓC(s+5/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
902500
= 22⋅54⋅192
|
Sign: |
1
|
Analytic conductor: |
23214.9 |
Root analytic conductor: |
12.3436 |
Motivic weight: |
5 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
2
|
Selberg data: |
(4, 902500, ( :5/2,5/2), 1)
|
Particular Values
L(3) |
= |
0 |
L(21) |
= |
0 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | C1 | (1−p2T)2 |
| 5 | | 1 |
| 19 | C1 | (1−p2T)2 |
good | 3 | D4 | 1+pT+128T2+p6T3+p10T4 |
| 7 | D4 | 1+114T+31099T2+114p5T3+p10T4 |
| 11 | D4 | 1−661T+430972T2−661p5T3+p10T4 |
| 13 | D4 | 1+1613T+1384022T2+1613p5T3+p10T4 |
| 17 | D4 | 1+64T+2804713T2+64p5T3+p10T4 |
| 23 | D4 | 1−3185T+14543782T2−3185p5T3+p10T4 |
| 29 | D4 | 1+2481T+6815332T2+2481p5T3+p10T4 |
| 31 | D4 | 1+1180T+21633278T2+1180p5T3+p10T4 |
| 37 | D4 | 1+10488T+155529814T2+10488p5T3+p10T4 |
| 41 | D4 | 1−16630T+170295586T2−16630p5T3+p10T4 |
| 43 | D4 | 1+11303T+297510638T2+11303p5T3+p10T4 |
| 47 | D4 | 1−12155T+47754214T2−12155p5T3+p10T4 |
| 53 | D4 | 1+20585T+812203882T2+20585p5T3+p10T4 |
| 59 | D4 | 1+78581T+2971672216T2+78581p5T3+p10T4 |
| 61 | D4 | 1−43621T+1919695356T2−43621p5T3+p10T4 |
| 67 | D4 | 1+7805T−748756690T2+7805p5T3+p10T4 |
| 71 | D4 | 1+62488T+4005427642T2+62488p5T3+p10T4 |
| 73 | D4 | 1+16218T+1004140843T2+16218p5T3+p10T4 |
| 79 | D4 | 1−67122T+7273984870T2−67122p5T3+p10T4 |
| 83 | D4 | 1−10714T+5751433246T2−10714p5T3+p10T4 |
| 89 | D4 | 1−128188T+8689285330T2−128188p5T3+p10T4 |
| 97 | D4 | 1+178558T+22668743394T2+178558p5T3+p10T4 |
show more | | |
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L(s)=p∏ j=1∏4(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−9.129434144609116193537200684772, −9.030320106263371284530200187759, −7.926825121170063782183372099451, −7.69905186000330746030367434424, −7.14630541703623830423773226349, −6.89283237113185984173114475938, −6.41566966873357572970608230544, −6.18982834416182642654956277109, −5.32681871822788878088288475727, −5.29162062857896153583668196511, −4.71336109804064401555062674096, −4.28163452685650426557895937720, −3.72595559469754306097378865617, −3.20689990266173227905758790300, −2.83144276114041174353568395192, −2.39867039197355751635816936463, −1.62859656729032328239270440338, −1.16302896072277631483145537948, 0, 0,
1.16302896072277631483145537948, 1.62859656729032328239270440338, 2.39867039197355751635816936463, 2.83144276114041174353568395192, 3.20689990266173227905758790300, 3.72595559469754306097378865617, 4.28163452685650426557895937720, 4.71336109804064401555062674096, 5.29162062857896153583668196511, 5.32681871822788878088288475727, 6.18982834416182642654956277109, 6.41566966873357572970608230544, 6.89283237113185984173114475938, 7.14630541703623830423773226349, 7.69905186000330746030367434424, 7.926825121170063782183372099451, 9.030320106263371284530200187759, 9.129434144609116193537200684772