L(s) = 1 | + 1.46i·5-s + (−0.866 + 0.5i)7-s + (−2.38 − 1.37i)11-s + (3.56 + 0.531i)13-s + (−2.39 − 4.15i)17-s + (−1.70 + 0.985i)19-s + (1.46 − 2.54i)23-s + 2.85·25-s + (−2.73 + 4.73i)29-s + 3.90i·31-s + (−0.733 − 1.26i)35-s + (1.87 + 1.08i)37-s + (8.18 + 4.72i)41-s + (5.80 + 10.0i)43-s + 10.3i·47-s + ⋯ |
L(s) = 1 | + 0.655i·5-s + (−0.327 + 0.188i)7-s + (−0.718 − 0.414i)11-s + (0.989 + 0.147i)13-s + (−0.582 − 1.00i)17-s + (−0.391 + 0.226i)19-s + (0.306 − 0.530i)23-s + 0.570·25-s + (−0.507 + 0.879i)29-s + 0.700i·31-s + (−0.123 − 0.214i)35-s + (0.308 + 0.178i)37-s + (1.27 + 0.737i)41-s + (0.885 + 1.53i)43-s + 1.51i·47-s + ⋯ |
Λ(s)=(=(3276s/2ΓC(s)L(s)(−0.378−0.925i)Λ(2−s)
Λ(s)=(=(3276s/2ΓC(s+1/2)L(s)(−0.378−0.925i)Λ(1−s)
Degree: |
2 |
Conductor: |
3276
= 22⋅32⋅7⋅13
|
Sign: |
−0.378−0.925i
|
Analytic conductor: |
26.1589 |
Root analytic conductor: |
5.11458 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ3276(2773,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 3276, ( :1/2), −0.378−0.925i)
|
Particular Values
L(1) |
≈ |
1.161438428 |
L(21) |
≈ |
1.161438428 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
| 7 | 1+(0.866−0.5i)T |
| 13 | 1+(−3.56−0.531i)T |
good | 5 | 1−1.46iT−5T2 |
| 11 | 1+(2.38+1.37i)T+(5.5+9.52i)T2 |
| 17 | 1+(2.39+4.15i)T+(−8.5+14.7i)T2 |
| 19 | 1+(1.70−0.985i)T+(9.5−16.4i)T2 |
| 23 | 1+(−1.46+2.54i)T+(−11.5−19.9i)T2 |
| 29 | 1+(2.73−4.73i)T+(−14.5−25.1i)T2 |
| 31 | 1−3.90iT−31T2 |
| 37 | 1+(−1.87−1.08i)T+(18.5+32.0i)T2 |
| 41 | 1+(−8.18−4.72i)T+(20.5+35.5i)T2 |
| 43 | 1+(−5.80−10.0i)T+(−21.5+37.2i)T2 |
| 47 | 1−10.3iT−47T2 |
| 53 | 1+1.24T+53T2 |
| 59 | 1+(3.95−2.28i)T+(29.5−51.0i)T2 |
| 61 | 1+(4.19+7.27i)T+(−30.5+52.8i)T2 |
| 67 | 1+(3.25+1.87i)T+(33.5+58.0i)T2 |
| 71 | 1+(8.58−4.95i)T+(35.5−61.4i)T2 |
| 73 | 1+5.93iT−73T2 |
| 79 | 1+12.8T+79T2 |
| 83 | 1−0.811iT−83T2 |
| 89 | 1+(−8.38−4.83i)T+(44.5+77.0i)T2 |
| 97 | 1+(8.23−4.75i)T+(48.5−84.0i)T2 |
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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
−8.995350266734057407002171131420, −8.085809484800860141392277661104, −7.39129684463666566995775082474, −6.49413872011367769765292562187, −6.08813398017682716179804700863, −5.04713928489105187420946033434, −4.23481704709797503130528483120, −3.03731812278832047250876919517, −2.71215053159404284731625829394, −1.20332869589576936635470190648,
0.37625078260148654967251345728, 1.66935938057994066818196207646, 2.68313450375786750735177026297, 3.90296207928110072006215359191, 4.36808803087003196684841962014, 5.52447426511927613764613010924, 5.97522350671891982157732689048, 6.99650270430023907907683882437, 7.68300201746910111997818293582, 8.559642346249128106735540552189