L(s) = 1 | + 3-s + 2·5-s + 9-s + 2·15-s − 25-s + 27-s + 2·45-s + 10·49-s + 20·53-s − 75-s + 16·79-s + 81-s − 8·83-s − 8·107-s + 18·121-s − 12·125-s + 127-s + 131-s + 2·135-s + 137-s + 139-s + 10·147-s + 149-s + 151-s + 157-s + 20·159-s + 163-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.516·15-s − 1/5·25-s + 0.192·27-s + 0.298·45-s + 10/7·49-s + 2.74·53-s − 0.115·75-s + 1.80·79-s + 1/9·81-s − 0.878·83-s − 0.773·107-s + 1.63·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.172·135-s + 0.0854·137-s + 0.0848·139-s + 0.824·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 1.58·159-s + 0.0783·163-s + ⋯ |
Λ(s)=(=(345600s/2ΓC(s)2L(s)Λ(2−s)
Λ(s)=(=(345600s/2ΓC(s+1/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
345600
= 29⋅33⋅52
|
Sign: |
1
|
Analytic conductor: |
22.0357 |
Root analytic conductor: |
2.16661 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
yes
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 345600, ( :1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
2.752819254 |
L(21) |
≈ |
2.752819254 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 3 | C1 | 1−T |
| 5 | C2 | 1−2T+pT2 |
good | 7 | C22 | 1−10T2+p2T4 |
| 11 | C22 | 1−18T2+p2T4 |
| 13 | C2 | (1−2T+pT2)(1+2T+pT2) |
| 17 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 19 | C2 | (1−8T+pT2)(1+8T+pT2) |
| 23 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 29 | C22 | 1+6T2+p2T4 |
| 31 | C2 | (1+pT2)2 |
| 37 | C2 | (1−10T+pT2)(1+10T+pT2) |
| 41 | C2 | (1−2T+pT2)(1+2T+pT2) |
| 43 | C2 | (1−12T+pT2)(1+12T+pT2) |
| 47 | C2 | (1+pT2)2 |
| 53 | C2 | (1−10T+pT2)2 |
| 59 | C22 | 1−82T2+p2T4 |
| 61 | C2 | (1−2T+pT2)(1+2T+pT2) |
| 67 | C2 | (1−8T+pT2)(1+8T+pT2) |
| 71 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 73 | C22 | 1−130T2+p2T4 |
| 79 | C2 | (1−8T+pT2)2 |
| 83 | C2 | (1+4T+pT2)2 |
| 89 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 97 | C2 | (1−18T+pT2)(1+18T+pT2) |
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
−8.804232518711272914975833640674, −8.339729437170921920814564547387, −7.88588785443659664968877050779, −7.29400030072680013487098644789, −6.95462669840527913917302121355, −6.43993342010759876282540261240, −5.72009755866980912480255592736, −5.60378398018998027832781220942, −4.89860843972320382069008847636, −4.21523796332319060680877486200, −3.79343751691397527212267079179, −3.06378196949189061119331802386, −2.37025221506316933204103603555, −1.96518584136898307589002473995, −0.950269898347283123135741361773,
0.950269898347283123135741361773, 1.96518584136898307589002473995, 2.37025221506316933204103603555, 3.06378196949189061119331802386, 3.79343751691397527212267079179, 4.21523796332319060680877486200, 4.89860843972320382069008847636, 5.60378398018998027832781220942, 5.72009755866980912480255592736, 6.43993342010759876282540261240, 6.95462669840527913917302121355, 7.29400030072680013487098644789, 7.88588785443659664968877050779, 8.339729437170921920814564547387, 8.804232518711272914975833640674