L(s) = 1 | − 2-s + 4-s + 7-s − 8-s + 9-s − 4·11-s − 14-s + 16-s − 18-s + 4·22-s + 4·23-s − 2·25-s + 28-s + 4·29-s − 32-s + 36-s − 4·37-s − 12·43-s − 4·44-s − 4·46-s + 49-s + 2·50-s − 16·53-s − 56-s − 4·58-s + 63-s + 64-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.267·14-s + 1/4·16-s − 0.235·18-s + 0.852·22-s + 0.834·23-s − 2/5·25-s + 0.188·28-s + 0.742·29-s − 0.176·32-s + 1/6·36-s − 0.657·37-s − 1.82·43-s − 0.603·44-s − 0.589·46-s + 1/7·49-s + 0.282·50-s − 2.19·53-s − 0.133·56-s − 0.525·58-s + 0.125·63-s + 1/8·64-s + ⋯ |
Λ(s)=(=(395136s/2ΓC(s)2L(s)−Λ(2−s)
Λ(s)=(=(395136s/2ΓC(s+1/2)2L(s)−Λ(1−s)
Degree: |
4 |
Conductor: |
395136
= 27⋅32⋅73
|
Sign: |
−1
|
Analytic conductor: |
25.1942 |
Root analytic conductor: |
2.24039 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
yes
|
Self-dual: |
yes
|
Analytic rank: |
1
|
Selberg data: |
(4, 395136, ( :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 | C1 | 1+T |
| 3 | C1×C1 | (1−T)(1+T) |
| 7 | C1 | 1−T |
good | 5 | C22 | 1+2T2+p2T4 |
| 11 | C2×C2 | (1+pT2)(1+4T+pT2) |
| 13 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 17 | C22 | 1−26T2+p2T4 |
| 19 | C2 | (1−8T+pT2)(1+8T+pT2) |
| 23 | C2×C2 | (1−8T+pT2)(1+4T+pT2) |
| 29 | C2 | (1−2T+pT2)2 |
| 31 | C22 | 1−34T2+p2T4 |
| 37 | C2×C2 | (1−2T+pT2)(1+6T+pT2) |
| 41 | C22 | 1+70T2+p2T4 |
| 43 | C2×C2 | (1+4T+pT2)(1+8T+pT2) |
| 47 | C22 | 1+14T2+p2T4 |
| 53 | C2×C2 | (1+2T+pT2)(1+14T+pT2) |
| 59 | C22 | 1+70T2+p2T4 |
| 61 | C22 | 1+50T2+p2T4 |
| 67 | C2 | (1−12T+pT2)(1+12T+pT2) |
| 71 | C2×C2 | (1−8T+pT2)(1+pT2) |
| 73 | C2 | (1−10T+pT2)(1+10T+pT2) |
| 79 | C2×C2 | (1+pT2)(1+16T+pT2) |
| 83 | C22 | 1+38T2+p2T4 |
| 89 | C22 | 1+30T2+p2T4 |
| 97 | C22 | 1+102T2+p2T4 |
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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
−8.411521090255994624785953587423, −8.052477028057648756115602305972, −7.54628166700195695041542955996, −7.23098376625686350759207285220, −6.58757381477372958101054261593, −6.28921047893520197455827551076, −5.54286182184328238279245201745, −5.01012367862760280737746076876, −4.79245704236827232363207773370, −3.94307762210019179524961979588, −3.21664771874035875051559967518, −2.75886454611533471050190400956, −1.96516192096943474380630638016, −1.28353908819465834293012833127, 0,
1.28353908819465834293012833127, 1.96516192096943474380630638016, 2.75886454611533471050190400956, 3.21664771874035875051559967518, 3.94307762210019179524961979588, 4.79245704236827232363207773370, 5.01012367862760280737746076876, 5.54286182184328238279245201745, 6.28921047893520197455827551076, 6.58757381477372958101054261593, 7.23098376625686350759207285220, 7.54628166700195695041542955996, 8.052477028057648756115602305972, 8.411521090255994624785953587423