L(s) = 1 | + 2-s + 3-s − 4-s + 5-s + 6-s − 3·8-s − 2·9-s + 10-s − 12-s + 15-s − 16-s − 2·18-s − 20-s − 3·24-s + 25-s − 5·27-s + 30-s + 7·31-s + 5·32-s + 2·36-s + 9·37-s − 3·40-s + 3·41-s + 6·43-s − 2·45-s − 48-s + 11·49-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.06·8-s − 2/3·9-s + 0.316·10-s − 0.288·12-s + 0.258·15-s − 1/4·16-s − 0.471·18-s − 0.223·20-s − 0.612·24-s + 1/5·25-s − 0.962·27-s + 0.182·30-s + 1.25·31-s + 0.883·32-s + 1/3·36-s + 1.47·37-s − 0.474·40-s + 0.468·41-s + 0.914·43-s − 0.298·45-s − 0.144·48-s + 11/7·49-s + ⋯ |
Λ(s)=(=(216000s/2ΓC(s)2L(s)Λ(2−s)
Λ(s)=(=(216000s/2ΓC(s+1/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
216000
= 26⋅33⋅53
|
Sign: |
1
|
Analytic conductor: |
13.7723 |
Root analytic conductor: |
1.92642 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
yes
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 216000, ( :1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
2.372686542 |
L(21) |
≈ |
2.372686542 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | C2 | 1−T+pT2 |
| 3 | C2 | 1−T+pT2 |
| 5 | C1 | 1−T |
good | 7 | C2 | (1−5T+pT2)(1+5T+pT2) |
| 11 | C22 | 1−6T2+p2T4 |
| 13 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 17 | C22 | 1−7T2+p2T4 |
| 19 | C22 | 1−10T2+p2T4 |
| 23 | C22 | 1+12T2+p2T4 |
| 29 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 31 | C2×C2 | (1−5T+pT2)(1−2T+pT2) |
| 37 | C2×C2 | (1−8T+pT2)(1−T+pT2) |
| 41 | C2×C2 | (1−5T+pT2)(1+2T+pT2) |
| 43 | C2×C2 | (1−10T+pT2)(1+4T+pT2) |
| 47 | C22 | 1+6T2+p2T4 |
| 53 | C2×C2 | (1−5T+pT2)(1+10T+pT2) |
| 59 | C22 | 1+15T2+p2T4 |
| 61 | C2 | (1−12T+pT2)(1+12T+pT2) |
| 67 | C2×C2 | (1−15T+pT2)(1−9T+pT2) |
| 71 | C2×C2 | (1−T+pT2)(1+16T+pT2) |
| 73 | C22 | 1−140T2+p2T4 |
| 79 | C2×C2 | (1−10T+pT2)(1−4T+pT2) |
| 83 | C2×C2 | (1+6T+pT2)(1+9T+pT2) |
| 89 | C2×C2 | (1+T+pT2)(1+14T+pT2) |
| 97 | C22 | 1+16T2+p2T4 |
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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
−9.041495130659913291796609340518, −8.619379858860742014519478097760, −8.115254595515861403192492614635, −7.82067700085579504791403153644, −7.05437394898644771902617788780, −6.50305877557587626274725562445, −5.90107096445752491992628901869, −5.67717689057213501223549252868, −5.08566268154462671636883732769, −4.34086900423517697146667239338, −4.09772230576046651221897837482, −3.24142362144906189270559175901, −2.75913019632310416779262742028, −2.22449489479160615519463093887, −0.851968183094800304293292875653,
0.851968183094800304293292875653, 2.22449489479160615519463093887, 2.75913019632310416779262742028, 3.24142362144906189270559175901, 4.09772230576046651221897837482, 4.34086900423517697146667239338, 5.08566268154462671636883732769, 5.67717689057213501223549252868, 5.90107096445752491992628901869, 6.50305877557587626274725562445, 7.05437394898644771902617788780, 7.82067700085579504791403153644, 8.115254595515861403192492614635, 8.619379858860742014519478097760, 9.041495130659913291796609340518