L(s) = 1 | − 3i·3-s + 5i·7-s − 9·9-s − 14·11-s + i·13-s − 46i·17-s + 19·19-s + 15·21-s + 46i·23-s + 27i·27-s − 14·29-s − 133·31-s + 42i·33-s − 258i·37-s + 3·39-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 0.269i·7-s − 0.333·9-s − 0.383·11-s + 0.0213i·13-s − 0.656i·17-s + 0.229·19-s + 0.155·21-s + 0.417i·23-s + 0.192i·27-s − 0.0896·29-s − 0.770·31-s + 0.221i·33-s − 1.14i·37-s + 0.0123·39-s + ⋯ |
Λ(s)=(=(1200s/2ΓC(s)L(s)(0.447−0.894i)Λ(4−s)
Λ(s)=(=(1200s/2ΓC(s+3/2)L(s)(0.447−0.894i)Λ(1−s)
Degree: |
2 |
Conductor: |
1200
= 24⋅3⋅52
|
Sign: |
0.447−0.894i
|
Analytic conductor: |
70.8022 |
Root analytic conductor: |
8.41440 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1200(49,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1200, ( :3/2), 0.447−0.894i)
|
Particular Values
L(2) |
≈ |
1.245673852 |
L(21) |
≈ |
1.245673852 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1+3iT |
| 5 | 1 |
good | 7 | 1−5iT−343T2 |
| 11 | 1+14T+1.33e3T2 |
| 13 | 1−iT−2.19e3T2 |
| 17 | 1+46iT−4.91e3T2 |
| 19 | 1−19T+6.85e3T2 |
| 23 | 1−46iT−1.21e4T2 |
| 29 | 1+14T+2.43e4T2 |
| 31 | 1+133T+2.97e4T2 |
| 37 | 1+258iT−5.06e4T2 |
| 41 | 1−84T+6.89e4T2 |
| 43 | 1−167iT−7.95e4T2 |
| 47 | 1−410iT−1.03e5T2 |
| 53 | 1−456iT−1.48e5T2 |
| 59 | 1+194T+2.05e5T2 |
| 61 | 1+17T+2.26e5T2 |
| 67 | 1−653iT−3.00e5T2 |
| 71 | 1+828T+3.57e5T2 |
| 73 | 1−570iT−3.89e5T2 |
| 79 | 1+552T+4.93e5T2 |
| 83 | 1+142iT−5.71e5T2 |
| 89 | 1−1.10e3T+7.04e5T2 |
| 97 | 1+841iT−9.12e5T2 |
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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
−9.345351384744648606772442839705, −8.777134374669988223617351143887, −7.63425352151050579720158848178, −7.28703502884847903674895974005, −6.09493377238305412731079992222, −5.46025372065920046688909614585, −4.37812333010377834639422300518, −3.13964558403526928726489327418, −2.22803927767488163653189640446, −0.997453533672714072584271601255,
0.33226466449880231217591547039, 1.84938106257661067139180973486, 3.10458170886009886005081542092, 3.98829699584281758309677404845, 4.92064878766796327484429902369, 5.75688026059346262893877801932, 6.72043598227828869950645212363, 7.66770248606964398152314325281, 8.490078953783317211010443185000, 9.236608865731083763355389776747