L(s) = 1 | − 2-s + 3-s − 6-s − 7-s + 8-s + 9-s − 11-s − 13-s + 14-s − 16-s − 17-s − 18-s − 21-s + 22-s + 24-s + 25-s + 26-s + 27-s + 29-s − 33-s + 34-s − 39-s + 2·41-s + 42-s − 47-s − 48-s − 50-s + ⋯ |
L(s) = 1 | − 2-s + 3-s − 6-s − 7-s + 8-s + 9-s − 11-s − 13-s + 14-s − 16-s − 17-s − 18-s − 21-s + 22-s + 24-s + 25-s + 26-s + 27-s + 29-s − 33-s + 34-s − 39-s + 2·41-s + 42-s − 47-s − 48-s − 50-s + ⋯ |
Λ(s)=(=(87s/2ΓC(s)L(s)Λ(1−s)
Λ(s)=(=(87s/2ΓC(s)L(s)Λ(1−s)
Degree: |
2 |
Conductor: |
87
= 3⋅29
|
Sign: |
1
|
Analytic conductor: |
0.0434186 |
Root analytic conductor: |
0.208371 |
Motivic weight: |
0 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
χ87(86,⋅)
|
Primitive: |
yes
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(2, 87, ( :0), 1)
|
Particular Values
L(21) |
≈ |
0.3960826179 |
L(21) |
≈ |
0.3960826179 |
L(1) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1−T |
| 29 | 1−T |
good | 2 | 1+T+T2 |
| 5 | (1−T)(1+T) |
| 7 | 1+T+T2 |
| 11 | 1+T+T2 |
| 13 | 1+T+T2 |
| 17 | 1+T+T2 |
| 19 | (1−T)(1+T) |
| 23 | (1−T)(1+T) |
| 31 | (1−T)(1+T) |
| 37 | (1−T)(1+T) |
| 41 | (1−T)2 |
| 43 | (1−T)(1+T) |
| 47 | 1+T+T2 |
| 53 | (1−T)(1+T) |
| 59 | (1−T)(1+T) |
| 61 | (1−T)(1+T) |
| 67 | 1+T+T2 |
| 71 | (1−T)(1+T) |
| 73 | (1−T)(1+T) |
| 79 | (1−T)(1+T) |
| 83 | (1−T)(1+T) |
| 89 | 1+T+T2 |
| 97 | (1−T)(1+T) |
show more | |
show less | |
L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−14.45102409066779657045769585783, −13.32162862510749184354432270374, −12.66997768785203913828312408994, −10.65716487846662969380774941347, −9.805231985523799033951222248495, −9.004858706215995835885407289726, −7.939979829085170543073834513148, −6.89861709118401082750268414601, −4.60444267458741594170483479006, −2.68368590772453422884570720998,
2.68368590772453422884570720998, 4.60444267458741594170483479006, 6.89861709118401082750268414601, 7.939979829085170543073834513148, 9.004858706215995835885407289726, 9.805231985523799033951222248495, 10.65716487846662969380774941347, 12.66997768785203913828312408994, 13.32162862510749184354432270374, 14.45102409066779657045769585783