L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 2·9-s − 14-s + 16-s + 10·17-s − 2·18-s + 4·23-s − 2·25-s − 28-s + 8·31-s + 32-s + 10·34-s − 2·36-s + 2·41-s + 4·46-s − 8·47-s + 49-s − 2·50-s − 56-s + 8·62-s + 2·63-s + 64-s + 10·68-s − 20·71-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 2/3·9-s − 0.267·14-s + 1/4·16-s + 2.42·17-s − 0.471·18-s + 0.834·23-s − 2/5·25-s − 0.188·28-s + 1.43·31-s + 0.176·32-s + 1.71·34-s − 1/3·36-s + 0.312·41-s + 0.589·46-s − 1.16·47-s + 1/7·49-s − 0.282·50-s − 0.133·56-s + 1.01·62-s + 0.251·63-s + 1/8·64-s + 1.21·68-s − 2.37·71-s + ⋯ |
Λ(s)=(=(43904s/2ΓC(s)2L(s)Λ(2−s)
Λ(s)=(=(43904s/2ΓC(s+1/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
43904
= 27⋅73
|
Sign: |
1
|
Analytic conductor: |
2.79935 |
Root analytic conductor: |
1.29349 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
yes
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 43904, ( :1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
2.008628703 |
L(21) |
≈ |
2.008628703 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | C1 | 1−T |
| 7 | C1 | 1+T |
good | 3 | C2 | (1−2T+pT2)(1+2T+pT2) |
| 5 | C22 | 1+2T2+p2T4 |
| 11 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 13 | C22 | 1+6T2+p2T4 |
| 17 | C2×C2 | (1−6T+pT2)(1−4T+pT2) |
| 19 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 23 | C2×C2 | (1−8T+pT2)(1+4T+pT2) |
| 29 | C22 | 1−34T2+p2T4 |
| 31 | C2×C2 | (1−6T+pT2)(1−2T+pT2) |
| 37 | C22 | 1+30T2+p2T4 |
| 41 | C2×C2 | (1−2T+pT2)(1+pT2) |
| 43 | C22 | 1+34T2+p2T4 |
| 47 | C2×C2 | (1−2T+pT2)(1+10T+pT2) |
| 53 | C22 | 1+22T2+p2T4 |
| 59 | C22 | 1+2T2+p2T4 |
| 61 | C22 | 1+2T2+p2T4 |
| 67 | C22 | 1+6T2+p2T4 |
| 71 | C2×C2 | (1+8T+pT2)(1+12T+pT2) |
| 73 | C2×C2 | (1−4T+pT2)(1+10T+pT2) |
| 79 | C2×C2 | (1−8T+pT2)(1+4T+pT2) |
| 83 | C22 | 1−86T2+p2T4 |
| 89 | C2×C2 | (1−18T+pT2)(1−16T+pT2) |
| 97 | C2×C2 | (1−14T+pT2)(1+12T+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
−10.26198625768805273878719158597, −9.807283911451784848867871077047, −9.233870777277990810646909989035, −8.626454361062853678938537989862, −7.87947203505069177916060814553, −7.72017961903121285224118371713, −6.93295761301336712015981974235, −6.28456104405352899088791784904, −5.84236365237513916779554422457, −5.27743924328440857639324718167, −4.72166922153586628950292773141, −3.80670019043265087333146656334, −3.17442428449498352846146247269, −2.69616292609918723606429253208, −1.27833369925900360338012527167,
1.27833369925900360338012527167, 2.69616292609918723606429253208, 3.17442428449498352846146247269, 3.80670019043265087333146656334, 4.72166922153586628950292773141, 5.27743924328440857639324718167, 5.84236365237513916779554422457, 6.28456104405352899088791784904, 6.93295761301336712015981974235, 7.72017961903121285224118371713, 7.87947203505069177916060814553, 8.626454361062853678938537989862, 9.233870777277990810646909989035, 9.807283911451784848867871077047, 10.26198625768805273878719158597