L(s) = 1 | + 2-s − 2·3-s − 4-s + 5-s − 2·6-s − 3·8-s + 9-s + 10-s + 2·12-s − 3·13-s − 2·15-s − 16-s + 18-s − 20-s + 6·24-s + 25-s − 3·26-s + 4·27-s − 2·30-s − 5·31-s + 5·32-s − 36-s + 21·37-s + 6·39-s − 3·40-s − 12·41-s + 6·43-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.447·5-s − 0.816·6-s − 1.06·8-s + 1/3·9-s + 0.316·10-s + 0.577·12-s − 0.832·13-s − 0.516·15-s − 1/4·16-s + 0.235·18-s − 0.223·20-s + 1.22·24-s + 1/5·25-s − 0.588·26-s + 0.769·27-s − 0.365·30-s − 0.898·31-s + 0.883·32-s − 1/6·36-s + 3.45·37-s + 0.960·39-s − 0.474·40-s − 1.87·41-s + 0.914·43-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 | C2 | 1+2T+pT2 |
| 5 | C1 | 1−T |
good | 7 | C22 | 1−5T2+p2T4 |
| 11 | C22 | 1+15T2+p2T4 |
| 13 | C2×C2 | (1+pT2)(1+3T+pT2) |
| 17 | C22 | 1−25T2+p2T4 |
| 19 | C22 | 1−10T2+p2T4 |
| 23 | C22 | 1+6T2+p2T4 |
| 29 | C22 | 1+4T2+p2T4 |
| 31 | C2×C2 | (1+T+pT2)(1+4T+pT2) |
| 37 | C2 | (1−11T+pT2)(1−10T+pT2) |
| 41 | C2×C2 | (1+2T+pT2)(1+10T+pT2) |
| 43 | C2×C2 | (1−5T+pT2)(1−T+pT2) |
| 47 | C22 | 1−48T2+p2T4 |
| 53 | C2×C2 | (1−14T+pT2)(1+10T+pT2) |
| 59 | C22 | 1+66T2+p2T4 |
| 61 | C22 | 1+29T2+p2T4 |
| 67 | C2×C2 | (1+6T+pT2)(1+12T+pT2) |
| 71 | C2×C2 | (1+T+pT2)(1+8T+pT2) |
| 73 | C22 | 1+94T2+p2T4 |
| 79 | C2×C2 | (1−10T+pT2)(1−T+pT2) |
| 83 | C2×C2 | (1−9T+pT2)(1+6T+pT2) |
| 89 | C2×C2 | (1+8T+pT2)(1+13T+pT2) |
| 97 | C2 | (1−8T+pT2)(1+8T+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
−8.887315493381641812267299163801, −8.374423639377667927524653520276, −7.76808777023899919160207095193, −7.25501711906557838799046022847, −6.67280506232813047140348475762, −6.09074702637691696932449821103, −5.86161561269523103309967568139, −5.31293846655662422468106502640, −4.92577200972167713662162849286, −4.34846347650913340301442118410, −3.92732225976336392665964410174, −2.90913798276067540906908593947, −2.51938651871105125230691073633, −1.20070290550731468156386632464, 0,
1.20070290550731468156386632464, 2.51938651871105125230691073633, 2.90913798276067540906908593947, 3.92732225976336392665964410174, 4.34846347650913340301442118410, 4.92577200972167713662162849286, 5.31293846655662422468106502640, 5.86161561269523103309967568139, 6.09074702637691696932449821103, 6.67280506232813047140348475762, 7.25501711906557838799046022847, 7.76808777023899919160207095193, 8.374423639377667927524653520276, 8.887315493381641812267299163801