L(s) = 1 | + i·3-s + 1.38i·7-s − 9-s + 6.23·11-s + 3i·13-s − 3.38i·17-s − 19-s − 1.38·21-s + 6.70i·23-s − i·27-s + 4.23·29-s + 3.09·31-s + 6.23i·33-s − 5i·37-s − 3·39-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.522i·7-s − 0.333·9-s + 1.88·11-s + 0.832i·13-s − 0.820i·17-s − 0.229·19-s − 0.301·21-s + 1.39i·23-s − 0.192i·27-s + 0.786·29-s + 0.555·31-s + 1.08i·33-s − 0.821i·37-s − 0.480·39-s + ⋯ |
Λ(s)=(=(6000s/2ΓC(s)L(s)−iΛ(2−s)
Λ(s)=(=(6000s/2ΓC(s+1/2)L(s)−iΛ(1−s)
Degree: |
2 |
Conductor: |
6000
= 24⋅3⋅53
|
Sign: |
−i
|
Analytic conductor: |
47.9102 |
Root analytic conductor: |
6.92172 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ6000(1249,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 6000, ( :1/2), −i)
|
Particular Values
L(1) |
≈ |
2.194872107 |
L(21) |
≈ |
2.194872107 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1−iT |
| 5 | 1 |
good | 7 | 1−1.38iT−7T2 |
| 11 | 1−6.23T+11T2 |
| 13 | 1−3iT−13T2 |
| 17 | 1+3.38iT−17T2 |
| 19 | 1+T+19T2 |
| 23 | 1−6.70iT−23T2 |
| 29 | 1−4.23T+29T2 |
| 31 | 1−3.09T+31T2 |
| 37 | 1+5iT−37T2 |
| 41 | 1+4.14T+41T2 |
| 43 | 1−2.38iT−43T2 |
| 47 | 1−9.18iT−47T2 |
| 53 | 1+3.61iT−53T2 |
| 59 | 1−10.7T+59T2 |
| 61 | 1+5.09T+61T2 |
| 67 | 1+8iT−67T2 |
| 71 | 1+1.14T+71T2 |
| 73 | 1−9.14iT−73T2 |
| 79 | 1−2.76T+79T2 |
| 83 | 1−8.32iT−83T2 |
| 89 | 1−5.47T+89T2 |
| 97 | 1+9.56iT−97T2 |
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L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−8.503319696765131294705433582317, −7.47974732061360613195214676689, −6.73908267041762519781100356696, −6.18130475766725826402720921661, −5.36166389742219699620843249776, −4.52404177533123052086773549903, −3.92770556282587765399507964437, −3.14516942269596473678648841214, −2.08903039400334965441480885432, −1.09247224217868012486904249346,
0.66049535098858556224487652619, 1.43505483538762821959349875866, 2.48481502600940957506789644544, 3.52634199587743765933659664259, 4.13977230156081707623358722189, 4.98242306379367774270142354107, 6.06864532716992106797738150033, 6.51979606830412014128565081458, 7.03724175706636410580252218956, 7.952537303851671843929230123238