L(s) = 1 | − 17.4i·5-s − 2.99·7-s + 10.6i·11-s + 43.3i·13-s + 37.8·17-s − 79.8i·19-s + 191.·23-s − 178.·25-s + 138. i·29-s + 212.·31-s + 52.1i·35-s − 270. i·37-s + 441.·41-s + 64.1i·43-s + 436.·47-s + ⋯ |
L(s) = 1 | − 1.55i·5-s − 0.161·7-s + 0.291i·11-s + 0.924i·13-s + 0.540·17-s − 0.964i·19-s + 1.73·23-s − 1.43·25-s + 0.889i·29-s + 1.22·31-s + 0.251i·35-s − 1.20i·37-s + 1.68·41-s + 0.227i·43-s + 1.35·47-s + ⋯ |
Λ(s)=(=(1152s/2ΓC(s)L(s)iΛ(4−s)
Λ(s)=(=(1152s/2ΓC(s+3/2)L(s)iΛ(1−s)
Degree: |
2 |
Conductor: |
1152
= 27⋅32
|
Sign: |
i
|
Analytic conductor: |
67.9702 |
Root analytic conductor: |
8.24440 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1152(577,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1152, ( :3/2), i)
|
Particular Values
L(2) |
≈ |
2.024518303 |
L(21) |
≈ |
2.024518303 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
good | 5 | 1+17.4iT−125T2 |
| 7 | 1+2.99T+343T2 |
| 11 | 1−10.6iT−1.33e3T2 |
| 13 | 1−43.3iT−2.19e3T2 |
| 17 | 1−37.8T+4.91e3T2 |
| 19 | 1+79.8iT−6.85e3T2 |
| 23 | 1−191.T+1.21e4T2 |
| 29 | 1−138.iT−2.43e4T2 |
| 31 | 1−212.T+2.97e4T2 |
| 37 | 1+270.iT−5.06e4T2 |
| 41 | 1−441.T+6.89e4T2 |
| 43 | 1−64.1iT−7.95e4T2 |
| 47 | 1−436.T+1.03e5T2 |
| 53 | 1+278.iT−1.48e5T2 |
| 59 | 1+830.iT−2.05e5T2 |
| 61 | 1−724.iT−2.26e5T2 |
| 67 | 1+859.iT−3.00e5T2 |
| 71 | 1+681.T+3.57e5T2 |
| 73 | 1+785.T+3.89e5T2 |
| 79 | 1+1.01e3T+4.93e5T2 |
| 83 | 1−467.iT−5.71e5T2 |
| 89 | 1+510.T+7.04e5T2 |
| 97 | 1+234.T+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
−9.082317791964432618961436192461, −8.696897889485989905644804789777, −7.58020778845096596192826576124, −6.78865748945506162008002634882, −5.64391547531652107107325396367, −4.82315385099333061410254280144, −4.24454605072518016685984935119, −2.85243219494961954132908540998, −1.49508914497466477179276019444, −0.59200752991177752465189296636,
1.03831991580905578803413510145, 2.74695332104695814375032805201, 3.09723682831516419102028626658, 4.28852507159995542693777041533, 5.66678552318488415548793401747, 6.23351540323450038393346052382, 7.21511579718168299776845782335, 7.78261672797151904285233618058, 8.768573805642631131465774585551, 9.965612169736428387958457661649