L(s) = 1 | − 5·9-s − 10·13-s + 6·17-s − 10·25-s − 18·29-s − 4·37-s − 13·49-s + 6·53-s + 20·61-s − 14·73-s + 16·81-s − 24·89-s − 20·97-s − 36·101-s − 22·109-s + 12·113-s + 50·117-s + 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 30·153-s + 157-s + 163-s + ⋯ |
L(s) = 1 | − 5/3·9-s − 2.77·13-s + 1.45·17-s − 2·25-s − 3.34·29-s − 0.657·37-s − 1.85·49-s + 0.824·53-s + 2.56·61-s − 1.63·73-s + 16/9·81-s − 2.54·89-s − 2.03·97-s − 3.58·101-s − 2.10·109-s + 1.12·113-s + 4.62·117-s + 1.27·121-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 − 2.42·153-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
Λ(s)=(=(1478656s/2ΓC(s)2L(s)Λ(2−s)
Λ(s)=(=(1478656s/2ΓC(s+1/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
1478656
= 212⋅192
|
Sign: |
1
|
Analytic conductor: |
94.2803 |
Root analytic conductor: |
3.11605 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
2
|
Selberg data: |
(4, 1478656, ( :1/2,1/2), 1)
|
Particular Values
L(1) |
= |
0 |
L(21) |
= |
0 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 19 | C1×C1 | (1−T)(1+T) |
good | 3 | C2 | (1−T+pT2)(1+T+pT2) |
| 5 | C2 | (1+pT2)2 |
| 7 | C2 | (1−T+pT2)(1+T+pT2) |
| 11 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 13 | C2 | (1+5T+pT2)2 |
| 17 | C2 | (1−3T+pT2)2 |
| 23 | C2 | (1−3T+pT2)(1+3T+pT2) |
| 29 | C2 | (1+9T+pT2)2 |
| 31 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 37 | C2 | (1+2T+pT2)2 |
| 41 | C2 | (1+pT2)2 |
| 43 | C2 | (1−8T+pT2)(1+8T+pT2) |
| 47 | C2 | (1+pT2)2 |
| 53 | C2 | (1−3T+pT2)2 |
| 59 | C2 | (1−9T+pT2)(1+9T+pT2) |
| 61 | C2 | (1−10T+pT2)2 |
| 67 | C2 | (1−5T+pT2)(1+5T+pT2) |
| 71 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 73 | C2 | (1+7T+pT2)2 |
| 79 | C2 | (1−10T+pT2)(1+10T+pT2) |
| 83 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 89 | C2 | (1+12T+pT2)2 |
| 97 | C2 | (1+10T+pT2)2 |
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
−7.49822591496797438401690336538, −7.14449686564876148233582580493, −6.79687210361750407728091466084, −5.93366234401884381484558908154, −5.56559135650201084358170066267, −5.32600053322851104543262796765, −5.20686401818851827647471141246, −4.07138629192131545636061302538, −3.98931312476793193392332292270, −3.12424893607844884324162519175, −2.80468657266648836263703727877, −2.16004950277651645615210004057, −1.68171654176783246558951167350, 0, 0,
1.68171654176783246558951167350, 2.16004950277651645615210004057, 2.80468657266648836263703727877, 3.12424893607844884324162519175, 3.98931312476793193392332292270, 4.07138629192131545636061302538, 5.20686401818851827647471141246, 5.32600053322851104543262796765, 5.56559135650201084358170066267, 5.93366234401884381484558908154, 6.79687210361750407728091466084, 7.14449686564876148233582580493, 7.49822591496797438401690336538