L(s) = 1 | + 2-s + 3-s − 4-s + 5-s + 6-s − 3·8-s + 9-s + 10-s − 12-s + 15-s − 16-s + 18-s − 6·19-s − 20-s − 12·23-s − 3·24-s + 25-s + 27-s − 10·29-s + 30-s + 5·32-s − 36-s − 6·38-s − 3·40-s − 10·43-s + 45-s − 12·46-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 + 0.258·15-s − 1/4·16-s + 0.235·18-s − 1.37·19-s − 0.223·20-s − 2.50·23-s − 0.612·24-s + 1/5·25-s + 0.192·27-s − 1.85·29-s + 0.182·30-s + 0.883·32-s − 1/6·36-s − 0.973·38-s − 0.474·40-s − 1.52·43-s + 0.149·45-s − 1.76·46-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 | C2 | 1−T+pT2 |
| 3 | C1 | 1−T |
| 5 | C1 | 1−T |
good | 7 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 11 | C22 | 1+4T2+p2T4 |
| 13 | C22 | 1+4T2+p2T4 |
| 17 | C22 | 1+26T2+p2T4 |
| 19 | C2×C2 | (1+pT2)(1+6T+pT2) |
| 23 | C2 | (1+6T+pT2)2 |
| 29 | C2×C2 | (1+4T+pT2)(1+6T+pT2) |
| 31 | C2 | (1−10T+pT2)(1+10T+pT2) |
| 37 | C22 | 1+24T2+p2T4 |
| 41 | C22 | 1−30T2+p2T4 |
| 43 | C2×C2 | (1+2T+pT2)(1+8T+pT2) |
| 47 | C2×C2 | (1−8T+pT2)(1+pT2) |
| 53 | C2×C2 | (1−6T+pT2)(1+10T+pT2) |
| 59 | C22 | 1−32T2+p2T4 |
| 61 | C22 | 1−58T2+p2T4 |
| 67 | C2×C2 | (1−12T+pT2)(1+6T+pT2) |
| 71 | C2×C2 | (1+2T+pT2)(1+8T+pT2) |
| 73 | C2×C2 | (1−16T+pT2)(1+10T+pT2) |
| 79 | C22 | 1−94T2+p2T4 |
| 83 | C22 | 1−46T2+p2T4 |
| 89 | C22 | 1+74T2+p2T4 |
| 97 | C2×C2 | (1+10T+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
−8.841949077573873823775079160332, −8.310387580710565750282513871774, −8.008759299199905597127093432791, −7.45120708943997382225768151918, −6.67801778768693653434556586990, −6.35074992708740092614652664848, −5.66703120125243249571427201763, −5.49998553895872731415445036150, −4.64979802762485321002813298462, −4.04402494779224559503968925162, −3.89398238296327538196455237767, −3.09438497492793793598861161837, −2.27164495510320738928114371029, −1.76895553779176625944886599601, 0,
1.76895553779176625944886599601, 2.27164495510320738928114371029, 3.09438497492793793598861161837, 3.89398238296327538196455237767, 4.04402494779224559503968925162, 4.64979802762485321002813298462, 5.49998553895872731415445036150, 5.66703120125243249571427201763, 6.35074992708740092614652664848, 6.67801778768693653434556586990, 7.45120708943997382225768151918, 8.008759299199905597127093432791, 8.310387580710565750282513871774, 8.841949077573873823775079160332