L(s) = 1 | − 2·2-s + 3-s + 2·4-s + 5-s − 2·6-s + 9-s − 2·10-s + 2·12-s + 15-s − 4·16-s − 2·18-s + 2·20-s + 25-s + 27-s − 2·30-s − 20·31-s + 8·32-s + 2·36-s + 45-s − 4·48-s − 10·49-s − 2·50-s − 20·53-s − 2·54-s + 2·60-s + 40·62-s − 8·64-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s + 1/3·9-s − 0.632·10-s + 0.577·12-s + 0.258·15-s − 16-s − 0.471·18-s + 0.447·20-s + 1/5·25-s + 0.192·27-s − 0.365·30-s − 3.59·31-s + 1.41·32-s + 1/3·36-s + 0.149·45-s − 0.577·48-s − 1.42·49-s − 0.282·50-s − 2.74·53-s − 0.272·54-s + 0.258·60-s + 5.08·62-s − 64-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+pT+pT2 |
| 3 | C1 | 1−T |
| 5 | C1 | 1−T |
good | 7 | C2 | (1−2T+pT2)(1+2T+pT2) |
| 11 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 13 | C2 | (1+pT2)2 |
| 17 | C22 | 1+2T2+p2T4 |
| 19 | C22 | 1−22T2+p2T4 |
| 23 | C22 | 1−30T2+p2T4 |
| 29 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 31 | C2 | (1+10T+pT2)2 |
| 37 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 41 | C2 | (1−10T+pT2)(1+10T+pT2) |
| 43 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 47 | C22 | 1−78T2+p2T4 |
| 53 | C2 | (1+10T+pT2)2 |
| 59 | C2 | (1−8T+pT2)(1+8T+pT2) |
| 61 | C22 | 1−58T2+p2T4 |
| 67 | C2 | (1−12T+pT2)(1+12T+pT2) |
| 71 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 73 | C2 | (1−10T+pT2)(1+10T+pT2) |
| 79 | C2 | (1+14T+pT2)2 |
| 83 | C2 | (1+pT2)2 |
| 89 | C2 | (1−14T+pT2)(1+14T+pT2) |
| 97 | C2 | (1−10T+pT2)(1+10T+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
−8.777463202787256456016698078145, −8.568121348401068092389681947177, −7.85126782769779844678487827092, −7.55865663172740494140791662232, −7.14925418958104587435907867580, −6.60444979582363780589168413242, −6.03848785485222247521986180613, −5.39414327193829845230627529171, −4.83299639814930091125721935543, −4.14575433902751758600957781044, −3.43838310204251765276920180681, −2.80479014504434913744491313053, −1.80502156484415218287852213649, −1.59243496351402759769462736519, 0,
1.59243496351402759769462736519, 1.80502156484415218287852213649, 2.80479014504434913744491313053, 3.43838310204251765276920180681, 4.14575433902751758600957781044, 4.83299639814930091125721935543, 5.39414327193829845230627529171, 6.03848785485222247521986180613, 6.60444979582363780589168413242, 7.14925418958104587435907867580, 7.55865663172740494140791662232, 7.85126782769779844678487827092, 8.568121348401068092389681947177, 8.777463202787256456016698078145