L(s) = 1 | + 5·9-s + 10·11-s − 10·19-s − 8·29-s + 20·31-s + 10·41-s + 10·49-s − 20·61-s − 20·79-s + 16·81-s + 18·89-s + 50·99-s + 4·101-s − 20·109-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s − 50·171-s + 173-s + ⋯ |
L(s) = 1 | + 5/3·9-s + 3.01·11-s − 2.29·19-s − 1.48·29-s + 3.59·31-s + 1.56·41-s + 10/7·49-s − 2.56·61-s − 2.25·79-s + 16/9·81-s + 1.90·89-s + 5.02·99-s + 0.398·101-s − 1.91·109-s + 4.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s − 3.82·171-s + 0.0760·173-s + ⋯ |
Λ(s)=(=(640000s/2ΓC(s)2L(s)Λ(2−s)
Λ(s)=(=(640000s/2ΓC(s+1/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
640000
= 210⋅54
|
Sign: |
1
|
Analytic conductor: |
40.8069 |
Root analytic conductor: |
2.52745 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 640000, ( :1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
2.929433293 |
L(21) |
≈ |
2.929433293 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 5 | | 1 |
good | 3 | C22 | 1−5T2+p2T4 |
| 7 | C22 | 1−10T2+p2T4 |
| 11 | C2 | (1−5T+pT2)2 |
| 13 | C2 | (1−pT2)2 |
| 17 | C22 | 1−9T2+p2T4 |
| 19 | C2 | (1+5T+pT2)2 |
| 23 | C22 | 1−10T2+p2T4 |
| 29 | C2 | (1+4T+pT2)2 |
| 31 | C2 | (1−10T+pT2)2 |
| 37 | C22 | 1+26T2+p2T4 |
| 41 | C2 | (1−5T+pT2)2 |
| 43 | C22 | 1−70T2+p2T4 |
| 47 | C22 | 1−30T2+p2T4 |
| 53 | C22 | 1−6T2+p2T4 |
| 59 | C2 | (1+pT2)2 |
| 61 | C2 | (1+10T+pT2)2 |
| 67 | C22 | 1−125T2+p2T4 |
| 71 | C2 | (1+pT2)2 |
| 73 | C22 | 1−121T2+p2T4 |
| 79 | C2 | (1+10T+pT2)2 |
| 83 | C22 | 1−165T2+p2T4 |
| 89 | C2 | (1−9T+pT2)2 |
| 97 | C22 | 1−94T2+p2T4 |
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
−10.31594868252216130527303204859, −10.24336329187136859823214909874, −9.443304060276063669156847471370, −9.182548237925223439875852383764, −9.052098412177689760219785597876, −8.395686243995346614595435488222, −7.917915564621921843013241512362, −7.41768024801698534526072636162, −6.83354096546273523050079644362, −6.63334225494251240614532572974, −6.08985683562610586151168257123, −6.07247840142845037611962991193, −4.87155616406333745522381581413, −4.29626325416612560031948243571, −4.14482125373430879833007252499, −3.98217840737894371696952159617, −2.98432329207846436430120275537, −2.15556571619399730829542466434, −1.49578044929089267131549979154, −1.00077759881575154699337302663,
1.00077759881575154699337302663, 1.49578044929089267131549979154, 2.15556571619399730829542466434, 2.98432329207846436430120275537, 3.98217840737894371696952159617, 4.14482125373430879833007252499, 4.29626325416612560031948243571, 4.87155616406333745522381581413, 6.07247840142845037611962991193, 6.08985683562610586151168257123, 6.63334225494251240614532572974, 6.83354096546273523050079644362, 7.41768024801698534526072636162, 7.917915564621921843013241512362, 8.395686243995346614595435488222, 9.052098412177689760219785597876, 9.182548237925223439875852383764, 9.443304060276063669156847471370, 10.24336329187136859823214909874, 10.31594868252216130527303204859