L(s) = 1 | − 2-s − 3-s − 4-s − 5-s + 6-s + 3·8-s + 9-s + 10-s + 12-s + 6·13-s + 15-s − 16-s − 18-s + 20-s − 3·24-s + 25-s − 6·26-s − 27-s − 30-s + 16·31-s − 5·32-s − 36-s − 6·37-s − 6·39-s − 3·40-s + 6·41-s − 45-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 + 1/3·9-s + 0.316·10-s + 0.288·12-s + 1.66·13-s + 0.258·15-s − 1/4·16-s − 0.235·18-s + 0.223·20-s − 0.612·24-s + 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.182·30-s + 2.87·31-s − 0.883·32-s − 1/6·36-s − 0.986·37-s − 0.960·39-s − 0.474·40-s + 0.937·41-s − 0.149·45-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) |
≈ |
0.8127848591 |
L(21) |
≈ |
0.8127848591 |
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 | C1 | 1+T |
| 5 | C1 | 1+T |
good | 7 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 11 | C22 | 1−6T2+p2T4 |
| 13 | C2×C2 | (1−6T+pT2)(1+pT2) |
| 17 | C22 | 1+2T2+p2T4 |
| 19 | C22 | 1−10T2+p2T4 |
| 23 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 29 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 31 | C2 | (1−8T+pT2)2 |
| 37 | C2×C2 | (1+2T+pT2)(1+4T+pT2) |
| 41 | C2×C2 | (1−4T+pT2)(1−2T+pT2) |
| 43 | C2 | (1−2T+pT2)(1+2T+pT2) |
| 47 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 53 | C2 | (1+2T+pT2)2 |
| 59 | C22 | 1−66T2+p2T4 |
| 61 | C22 | 1+50T2+p2T4 |
| 67 | C2×C2 | (1−12T+pT2)(1+pT2) |
| 71 | C2×C2 | (1−10T+pT2)(1−2T+pT2) |
| 73 | C22 | 1+94T2+p2T4 |
| 79 | C2×C2 | (1−4T+pT2)(1+8T+pT2) |
| 83 | C2×C2 | (1+pT2)(1+12T+pT2) |
| 89 | C2×C2 | (1−14T+pT2)(1+8T+pT2) |
| 97 | C22 | 1−74T2+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
−8.905977212366117482335025310366, −8.566266728237752344812164323656, −8.164784813489564550585907462237, −7.85923543569937005011130177851, −7.14084006027180754987274182931, −6.68518872303551465466396243472, −6.16695895670544286351716080604, −5.71340205752869075700678895457, −4.93302791533296556425176498902, −4.60404315210379879095503014039, −3.91910375636700705199954650109, −3.52181352865394515288844372408, −2.54400172959357573602668819491, −1.39941922193662355717119149365, −0.75115206295181260150647173214,
0.75115206295181260150647173214, 1.39941922193662355717119149365, 2.54400172959357573602668819491, 3.52181352865394515288844372408, 3.91910375636700705199954650109, 4.60404315210379879095503014039, 4.93302791533296556425176498902, 5.71340205752869075700678895457, 6.16695895670544286351716080604, 6.68518872303551465466396243472, 7.14084006027180754987274182931, 7.85923543569937005011130177851, 8.164784813489564550585907462237, 8.566266728237752344812164323656, 8.905977212366117482335025310366