L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 6·5-s − 6·6-s − 7·7-s + 16·8-s + 6·9-s − 12·10-s − 10·11-s − 12·12-s − 14·14-s + 18·15-s + 32·16-s − 12·17-s + 12·18-s + 57·19-s − 24·20-s + 21·21-s − 20·22-s − 40·23-s − 48·24-s − 25-s − 9·27-s − 28·28-s + 32·29-s + 36·30-s + ⋯ |
L(s) = 1 | + 2-s − 3-s + 4-s − 6/5·5-s − 6-s − 7-s + 2·8-s + 2/3·9-s − 6/5·10-s − 0.909·11-s − 12-s − 14-s + 6/5·15-s + 2·16-s − 0.705·17-s + 2/3·18-s + 3·19-s − 6/5·20-s + 21-s − 0.909·22-s − 1.73·23-s − 2·24-s − 0.0399·25-s − 1/3·27-s − 28-s + 1.10·29-s + 6/5·30-s + ⋯ |
Λ(s)=(=(441s/2ΓC(s)2L(s)Λ(3−s)
Λ(s)=(=(441s/2ΓC(s+1)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
441
= 32⋅72
|
Sign: |
1
|
Analytic conductor: |
0.327422 |
Root analytic conductor: |
0.756444 |
Motivic weight: |
2 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 441, ( :1,1), 1)
|
Particular Values
L(23) |
≈ |
0.8839576896 |
L(21) |
≈ |
0.8839576896 |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 3 | C2 | 1+pT+pT2 |
| 7 | C2 | 1+pT+p2T2 |
good | 2 | C1×C2 | (1−pT)2(1+pT+p2T2) |
| 5 | C22 | 1+6T+37T2+6p2T3+p4T4 |
| 11 | C22 | 1+10T−21T2+10p2T3+p4T4 |
| 13 | C2 | (1−23T+p2T2)(1+23T+p2T2) |
| 17 | C22 | 1+12T+337T2+12p2T3+p4T4 |
| 19 | C1×C2 | (1−pT)2(1−pT+p2T2) |
| 23 | C22 | 1+40T+1071T2+40p2T3+p4T4 |
| 29 | C2 | (1−16T+p2T2)2 |
| 31 | C22 | 1−9T+988T2−9p2T3+p4T4 |
| 37 | C22 | 1+5T−1344T2+5p2T3+p4T4 |
| 41 | C22 | 1−2774T2+p4T4 |
| 43 | C2 | (1+19T+p2T2)2 |
| 47 | C22 | 1+90T+4909T2+90p2T3+p4T4 |
| 53 | C22 | 1−32T−1785T2−32p2T3+p4T4 |
| 59 | C22 | 1−72T+5209T2−72p2T3+p4T4 |
| 61 | C22 | 1−36T+4153T2−36p2T3+p4T4 |
| 67 | C22 | 1+59T−1008T2+59p2T3+p4T4 |
| 71 | C2 | (1+26T+p2T2)2 |
| 73 | C22 | 1+33T+5692T2+33p2T3+p4T4 |
| 79 | C22 | 1+47T−4032T2+47p2T3+p4T4 |
| 83 | C22 | 1−13190T2+p4T4 |
| 89 | C22 | 1−204T+21793T2−204p2T3+p4T4 |
| 97 | C22 | 1−16466T2+p4T4 |
show more | | |
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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
−17.98575500164893497617612111575, −17.90683416423905283373161651990, −16.46597840674563552948280763401, −16.28447276859396086641175184325, −15.75797551673013038124459084352, −15.63988802368905518660367202192, −14.34528053835078955911414753800, −13.54432850756280588224298023507, −13.22209945964985216508732192582, −12.27880611898751037709276697879, −11.55461121444636327817579051007, −11.53885667903414972455922494576, −10.13928041292070285369187287890, −10.06348780879576766014782917988, −8.017509227007492090205306035738, −7.47388105315423938103714340800, −6.62750552293455248086161610210, −5.50551512778034746268598153449, −4.61368157550132321146797149827, −3.43300208294605646978948795414,
3.43300208294605646978948795414, 4.61368157550132321146797149827, 5.50551512778034746268598153449, 6.62750552293455248086161610210, 7.47388105315423938103714340800, 8.017509227007492090205306035738, 10.06348780879576766014782917988, 10.13928041292070285369187287890, 11.53885667903414972455922494576, 11.55461121444636327817579051007, 12.27880611898751037709276697879, 13.22209945964985216508732192582, 13.54432850756280588224298023507, 14.34528053835078955911414753800, 15.63988802368905518660367202192, 15.75797551673013038124459084352, 16.28447276859396086641175184325, 16.46597840674563552948280763401, 17.90683416423905283373161651990, 17.98575500164893497617612111575