L(s) = 1 | − i·3-s − 2.23i·5-s + 1.23i·7-s − 9-s − 2·11-s − 5.23i·13-s − 2.23·15-s + i·17-s − 6.47·19-s + 1.23·21-s + 4i·23-s − 5.00·25-s + i·27-s + 4·29-s − 2.47·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.999i·5-s + 0.467i·7-s − 0.333·9-s − 0.603·11-s − 1.45i·13-s − 0.577·15-s + 0.242i·17-s − 1.48·19-s + 0.269·21-s + 0.834i·23-s − 1.00·25-s + 0.192i·27-s + 0.742·29-s − 0.444·31-s + ⋯ |
Λ(s)=(=(4080s/2ΓC(s)L(s)−iΛ(2−s)
Λ(s)=(=(4080s/2ΓC(s+1/2)L(s)−iΛ(1−s)
Degree: |
2 |
Conductor: |
4080
= 24⋅3⋅5⋅17
|
Sign: |
−i
|
Analytic conductor: |
32.5789 |
Root analytic conductor: |
5.70779 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ4080(2449,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 4080, ( :1/2), −i)
|
Particular Values
L(1) |
≈ |
0.2819479959 |
L(21) |
≈ |
0.2819479959 |
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+2.23iT |
| 17 | 1−iT |
good | 7 | 1−1.23iT−7T2 |
| 11 | 1+2T+11T2 |
| 13 | 1+5.23iT−13T2 |
| 19 | 1+6.47T+19T2 |
| 23 | 1−4iT−23T2 |
| 29 | 1−4T+29T2 |
| 31 | 1+2.47T+31T2 |
| 37 | 1−2iT−37T2 |
| 41 | 1+7.70T+41T2 |
| 43 | 1−12.1iT−43T2 |
| 47 | 1+10.4iT−47T2 |
| 53 | 1−10.9iT−53T2 |
| 59 | 1−11.7T+59T2 |
| 61 | 1−4.47T+61T2 |
| 67 | 1−1.70iT−67T2 |
| 71 | 1+11.2T+71T2 |
| 73 | 1+8.76iT−73T2 |
| 79 | 1−0.944T+79T2 |
| 83 | 1−10.4iT−83T2 |
| 89 | 1+12.4T+89T2 |
| 97 | 1+1.70iT−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
−8.356268082747924537089856823767, −8.154217623579506587427544614520, −7.27401757610306078317800045097, −6.28968987205691368975934624584, −5.59072527763094925640632060388, −5.09300375656129953604384451321, −4.10610048722845622415751108265, −3.03120219789291241890020678953, −2.15056542184496250130093119113, −1.10301304985688485936536160564,
0.082846234740168725565072584165, 1.98247643508177530252271915590, 2.67782010055908602708073282445, 3.79966083414469113715176292498, 4.27089181925853376177795912031, 5.17843912155961614641938952091, 6.19242628248492608153194271865, 6.81071349077508861342855292016, 7.32066156544698011912306538269, 8.434058776989840827254572558587