L(s) = 1 | + 6i·5-s − 21.1·7-s − 42.3i·11-s + 20i·13-s + 8·17-s − 84.6i·19-s + 169.·23-s + 89·25-s + 46i·29-s + 21.1·31-s − 126. i·35-s + 164i·37-s − 312·41-s + 423. i·43-s − 169.·47-s + ⋯ |
L(s) = 1 | + 0.536i·5-s − 1.14·7-s − 1.16i·11-s + 0.426i·13-s + 0.114·17-s − 1.02i·19-s + 1.53·23-s + 0.711·25-s + 0.294i·29-s + 0.122·31-s − 0.613i·35-s + 0.728i·37-s − 1.18·41-s + 1.50i·43-s − 0.525·47-s + ⋯ |
Λ(s)=(=(1152s/2ΓC(s)L(s)(−0.707−0.707i)Λ(4−s)
Λ(s)=(=(1152s/2ΓC(s+3/2)L(s)(−0.707−0.707i)Λ(1−s)
Degree: |
2 |
Conductor: |
1152
= 27⋅32
|
Sign: |
−0.707−0.707i
|
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), −0.707−0.707i)
|
Particular Values
L(2) |
≈ |
0.6691229540 |
L(21) |
≈ |
0.6691229540 |
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 | 1−6iT−125T2 |
| 7 | 1+21.1T+343T2 |
| 11 | 1+42.3iT−1.33e3T2 |
| 13 | 1−20iT−2.19e3T2 |
| 17 | 1−8T+4.91e3T2 |
| 19 | 1+84.6iT−6.85e3T2 |
| 23 | 1−169.T+1.21e4T2 |
| 29 | 1−46iT−2.43e4T2 |
| 31 | 1−21.1T+2.97e4T2 |
| 37 | 1−164iT−5.06e4T2 |
| 41 | 1+312T+6.89e4T2 |
| 43 | 1−423.iT−7.95e4T2 |
| 47 | 1+169.T+1.03e5T2 |
| 53 | 1−266iT−1.48e5T2 |
| 59 | 1−253.iT−2.05e5T2 |
| 61 | 1+132iT−2.26e5T2 |
| 67 | 1+507.iT−3.00e5T2 |
| 71 | 1+677.T+3.57e5T2 |
| 73 | 1+246T+3.89e5T2 |
| 79 | 1+232.T+4.93e5T2 |
| 83 | 1−973.iT−5.71e5T2 |
| 89 | 1+1.39e3T+7.04e5T2 |
| 97 | 1+302T+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
−9.612584530081858222026007341083, −9.028766306522555490089514479270, −8.180281603052474915530171895011, −6.82726479786059202067521158133, −6.70452409333778366591594433586, −5.59942407148581592771416956372, −4.54520414980913956935308001048, −3.11875767581333603192759740542, −2.96907258457680943611021892785, −1.10105881699621432751345220702,
0.17824465339069045753305772108, 1.50701856831571009167156852478, 2.82864035433359919397687291548, 3.77781344438274703600685602465, 4.85134172251848842998253374235, 5.65347616855220775776091121568, 6.73447169633712631361350903956, 7.30918388910267536820033015636, 8.423770315352353201425632992008, 9.155976428761821389169852970025