L(s) = 1 | + 2-s + 3-s − 4-s − 5-s + 6-s − 3·8-s + 9-s − 10-s − 12-s − 4·13-s − 15-s − 16-s − 4·17-s + 18-s + 8·19-s + 20-s − 3·24-s + 25-s − 4·26-s + 27-s + 4·29-s − 30-s + 5·32-s − 4·34-s − 36-s − 20·37-s + 8·38-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 − 1.10·13-s − 0.258·15-s − 1/4·16-s − 0.970·17-s + 0.235·18-s + 1.83·19-s + 0.223·20-s − 0.612·24-s + 1/5·25-s − 0.784·26-s + 0.192·27-s + 0.742·29-s − 0.182·30-s + 0.883·32-s − 0.685·34-s − 1/6·36-s − 3.28·37-s + 1.29·38-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: |
no
|
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+pT2)2 |
| 11 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 13 | C2 | (1+2T+pT2)2 |
| 17 | C2 | (1+2T+pT2)2 |
| 19 | C2 | (1−4T+pT2)2 |
| 23 | C2 | (1+pT2)2 |
| 29 | C2 | (1−2T+pT2)2 |
| 31 | C2 | (1+pT2)2 |
| 37 | C2 | (1+10T+pT2)2 |
| 41 | C2 | (1−10T+pT2)(1+10T+pT2) |
| 43 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 47 | C2 | (1−8T+pT2)(1+8T+pT2) |
| 53 | C2 | (1−10T+pT2)(1+10T+pT2) |
| 59 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 61 | C2 | (1−2T+pT2)(1+2T+pT2) |
| 67 | C2 | (1−12T+pT2)(1+12T+pT2) |
| 71 | C2 | (1−8T+pT2)2 |
| 73 | C2 | (1−10T+pT2)(1+10T+pT2) |
| 79 | C2 | (1+pT2)2 |
| 83 | C2 | (1+12T+pT2)2 |
| 89 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 97 | C2 | (1−2T+pT2)(1+2T+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
−8.603932276417102120989834189554, −8.554588868470699894187777077269, −7.84465462281701093084420479537, −7.39546104131459972836674622338, −6.76955885158912340897756950945, −6.60942929553535216154216963636, −5.59929057414059862389948014289, −5.04860090093668311608273372623, −4.97287019204270070424993179755, −4.17881549969972779526027182436, −3.63412818236731287070311550143, −3.10526976657196724130917180316, −2.58465419848810939203642390782, −1.51207711394055808445468187309, 0,
1.51207711394055808445468187309, 2.58465419848810939203642390782, 3.10526976657196724130917180316, 3.63412818236731287070311550143, 4.17881549969972779526027182436, 4.97287019204270070424993179755, 5.04860090093668311608273372623, 5.59929057414059862389948014289, 6.60942929553535216154216963636, 6.76955885158912340897756950945, 7.39546104131459972836674622338, 7.84465462281701093084420479537, 8.554588868470699894187777077269, 8.603932276417102120989834189554