L(s) = 1 | − 3-s − 4-s − 5-s − 5·7-s + 9-s + 12-s + 15-s + 16-s + 20-s + 5·21-s − 4·25-s − 27-s + 5·28-s + 5·35-s − 36-s − 5·37-s − 10·41-s − 15·43-s − 45-s + 47-s − 48-s + 18·49-s + 20·59-s − 60-s − 5·63-s − 64-s + 4·75-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1/2·4-s − 0.447·5-s − 1.88·7-s + 1/3·9-s + 0.288·12-s + 0.258·15-s + 1/4·16-s + 0.223·20-s + 1.09·21-s − 4/5·25-s − 0.192·27-s + 0.944·28-s + 0.845·35-s − 1/6·36-s − 0.821·37-s − 1.56·41-s − 2.28·43-s − 0.149·45-s + 0.145·47-s − 0.144·48-s + 18/7·49-s + 2.60·59-s − 0.129·60-s − 0.629·63-s − 1/8·64-s + 0.461·75-s + ⋯ |
Λ(s)=(=(132300s/2ΓC(s)2L(s)Λ(2−s)
Λ(s)=(=(132300s/2ΓC(s+1/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
132300
= 22⋅33⋅52⋅72
|
Sign: |
1
|
Analytic conductor: |
8.43556 |
Root analytic conductor: |
1.70423 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
yes
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 132300, ( :1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
0.4379027794 |
L(21) |
≈ |
0.4379027794 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | C2 | 1+T2 |
| 3 | C1 | 1+T |
| 5 | C2 | 1+T+pT2 |
| 7 | C2 | 1+5T+pT2 |
good | 11 | C22 | 1−9T2+p2T4 |
| 13 | C22 | 1+20T2+p2T4 |
| 17 | C2 | (1−5T+pT2)(1+5T+pT2) |
| 19 | C22 | 1−22T2+p2T4 |
| 23 | C22 | 1+4T2+p2T4 |
| 29 | C22 | 1−16T2+p2T4 |
| 31 | C22 | 1+32T2+p2T4 |
| 37 | C2×C2 | (1+T+pT2)(1+4T+pT2) |
| 41 | C2×C2 | (1+pT2)(1+10T+pT2) |
| 43 | C2×C2 | (1+4T+pT2)(1+11T+pT2) |
| 47 | C2×C2 | (1−8T+pT2)(1+7T+pT2) |
| 53 | C22 | 1+19T2+p2T4 |
| 59 | C2×C2 | (1−14T+pT2)(1−6T+pT2) |
| 61 | C22 | 1+79T2+p2T4 |
| 67 | C2 | (1−T+pT2)(1+T+pT2) |
| 71 | C2 | (1−8T+pT2)(1+8T+pT2) |
| 73 | C22 | 1+30T2+p2T4 |
| 79 | C2×C2 | (1−3T+pT2)(1+12T+pT2) |
| 83 | C2×C2 | (1−12T+pT2)(1−7T+pT2) |
| 89 | C2×C2 | (1−12T+pT2)(1+7T+pT2) |
| 97 | C22 | 1−50T2+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
−9.607208471564681200929939698031, −8.848816285099318986087744849626, −8.505698999019485589977521256618, −7.984567411096763444580986362622, −7.13682700831533312015440131557, −6.94316955660584393926016610480, −6.41578453280906087258831566326, −5.88874387429572538588576714330, −5.30963037576648192115197073588, −4.81816101464795294409726127939, −3.91421469398545160810260207237, −3.60792296384436066119156884197, −3.03950771406390117947679780329, −1.93991427194705757404501644184, −0.46134343672254373040015387019,
0.46134343672254373040015387019, 1.93991427194705757404501644184, 3.03950771406390117947679780329, 3.60792296384436066119156884197, 3.91421469398545160810260207237, 4.81816101464795294409726127939, 5.30963037576648192115197073588, 5.88874387429572538588576714330, 6.41578453280906087258831566326, 6.94316955660584393926016610480, 7.13682700831533312015440131557, 7.984567411096763444580986362622, 8.505698999019485589977521256618, 8.848816285099318986087744849626, 9.607208471564681200929939698031