L(s) = 1 | − 3-s − 5-s + 2·7-s + 9-s − 6·11-s + 15-s − 2·21-s + 25-s − 27-s + 6·33-s − 2·35-s + 8·43-s − 45-s − 2·49-s + 12·53-s + 6·55-s − 18·59-s + 4·61-s + 2·63-s − 4·67-s − 12·71-s − 75-s − 12·77-s + 81-s − 6·99-s − 10·103-s + 2·105-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.80·11-s + 0.258·15-s − 0.436·21-s + 1/5·25-s − 0.192·27-s + 1.04·33-s − 0.338·35-s + 1.21·43-s − 0.149·45-s − 2/7·49-s + 1.64·53-s + 0.809·55-s − 2.34·59-s + 0.512·61-s + 0.251·63-s − 0.488·67-s − 1.42·71-s − 0.115·75-s − 1.36·77-s + 1/9·81-s − 0.603·99-s − 0.985·103-s + 0.195·105-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: |
1
|
Selberg data: |
(4, 216000, ( :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 |
| 3 | C1 | 1+T |
| 5 | C1 | 1+T |
good | 7 | C2×C2 | (1−4T+pT2)(1+2T+pT2) |
| 11 | C2×C2 | (1+pT2)(1+6T+pT2) |
| 13 | C22 | 1−2T2+p2T4 |
| 17 | C2 | (1+pT2)2 |
| 19 | C22 | 1+10T2+p2T4 |
| 23 | C22 | 1−26T2+p2T4 |
| 29 | C22 | 1−50T2+p2T4 |
| 31 | C2 | (1−10T+pT2)(1+10T+pT2) |
| 37 | C22 | 1−2T2+p2T4 |
| 41 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 43 | C2 | (1−4T+pT2)2 |
| 47 | C22 | 1−14T2+p2T4 |
| 53 | C2 | (1−6T+pT2)2 |
| 59 | C2×C2 | (1+6T+pT2)(1+12T+pT2) |
| 61 | C2×C2 | (1−8T+pT2)(1+4T+pT2) |
| 67 | C2×C2 | (1−4T+pT2)(1+8T+pT2) |
| 71 | C2×C2 | (1+pT2)(1+12T+pT2) |
| 73 | C22 | 1−74T2+p2T4 |
| 79 | C2 | (1−10T+pT2)(1+10T+pT2) |
| 83 | C22 | 1−86T2+p2T4 |
| 89 | C22 | 1+70T2+p2T4 |
| 97 | C2 | (1−14T+pT2)(1+14T+pT2) |
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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
−8.683893751284429556749676941632, −8.242498904373773901863115668516, −7.83298189622501308320383508503, −7.38361830089163477437182996650, −7.11172911542884038573278201249, −6.25930713551989881226908856716, −5.78271512389619020996117690528, −5.34037661038038998021641847667, −4.79150066353033527558648259761, −4.43669898042292026640195142146, −3.70528607298888800971732556574, −2.86820919626464618858559542612, −2.32769520098898456110671665937, −1.29044932946073407894236523594, 0,
1.29044932946073407894236523594, 2.32769520098898456110671665937, 2.86820919626464618858559542612, 3.70528607298888800971732556574, 4.43669898042292026640195142146, 4.79150066353033527558648259761, 5.34037661038038998021641847667, 5.78271512389619020996117690528, 6.25930713551989881226908856716, 7.11172911542884038573278201249, 7.38361830089163477437182996650, 7.83298189622501308320383508503, 8.242498904373773901863115668516, 8.683893751284429556749676941632