L(s) = 1 | + (−0.866 + 1.5i)2-s + (−0.5 − 0.866i)4-s + (−1.73 − 3i)5-s + (0.5 − 0.866i)7-s − 1.73·8-s + 6·10-s + (1.73 − 3i)11-s + (−2.5 − 4.33i)13-s + (0.866 + 1.5i)14-s + (2.49 − 4.33i)16-s − 19-s + (−1.73 + 3i)20-s + (3 + 5.19i)22-s + (3.46 + 6i)23-s + (−3.5 + 6.06i)25-s + 8.66·26-s + ⋯ |
L(s) = 1 | + (−0.612 + 1.06i)2-s + (−0.250 − 0.433i)4-s + (−0.774 − 1.34i)5-s + (0.188 − 0.327i)7-s − 0.612·8-s + 1.89·10-s + (0.522 − 0.904i)11-s + (−0.693 − 1.20i)13-s + (0.231 + 0.400i)14-s + (0.624 − 1.08i)16-s − 0.229·19-s + (−0.387 + 0.670i)20-s + (0.639 + 1.10i)22-s + (0.722 + 1.25i)23-s + (−0.700 + 1.21i)25-s + 1.69·26-s + ⋯ |
Λ(s)=(=(243s/2ΓC(s)L(s)(0.766+0.642i)Λ(2−s)
Λ(s)=(=(243s/2ΓC(s+1/2)L(s)(0.766+0.642i)Λ(1−s)
Degree: |
2 |
Conductor: |
243
= 35
|
Sign: |
0.766+0.642i
|
Analytic conductor: |
1.94036 |
Root analytic conductor: |
1.39296 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ243(163,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 243, ( :1/2), 0.766+0.642i)
|
Particular Values
L(1) |
≈ |
0.598307−0.217766i |
L(21) |
≈ |
0.598307−0.217766i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
good | 2 | 1+(0.866−1.5i)T+(−1−1.73i)T2 |
| 5 | 1+(1.73+3i)T+(−2.5+4.33i)T2 |
| 7 | 1+(−0.5+0.866i)T+(−3.5−6.06i)T2 |
| 11 | 1+(−1.73+3i)T+(−5.5−9.52i)T2 |
| 13 | 1+(2.5+4.33i)T+(−6.5+11.2i)T2 |
| 17 | 1+17T2 |
| 19 | 1+T+19T2 |
| 23 | 1+(−3.46−6i)T+(−11.5+19.9i)T2 |
| 29 | 1+(−1.73+3i)T+(−14.5−25.1i)T2 |
| 31 | 1+(2.5+4.33i)T+(−15.5+26.8i)T2 |
| 37 | 1+T+37T2 |
| 41 | 1+(1.73+3i)T+(−20.5+35.5i)T2 |
| 43 | 1+(−0.5+0.866i)T+(−21.5−37.2i)T2 |
| 47 | 1+(−1.73+3i)T+(−23.5−40.7i)T2 |
| 53 | 1+10.3T+53T2 |
| 59 | 1+(1.73+3i)T+(−29.5+51.0i)T2 |
| 61 | 1+(1−1.73i)T+(−30.5−52.8i)T2 |
| 67 | 1+(4+6.92i)T+(−33.5+58.0i)T2 |
| 71 | 1−10.3T+71T2 |
| 73 | 1−2T+73T2 |
| 79 | 1+(−0.5+0.866i)T+(−39.5−68.4i)T2 |
| 83 | 1+(3.46−6i)T+(−41.5−71.8i)T2 |
| 89 | 1−10.3T+89T2 |
| 97 | 1+(8.5−14.7i)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
−12.07782790986596153554552766680, −11.11878037601829153735185745971, −9.616641942505299831624556621184, −8.789323586165361484461357426998, −8.006903235802678557635785973711, −7.39680556639963975985484355977, −5.93982797556827404509640005013, −4.96584975226237627380466797561, −3.49625084704881218547416639341, −0.62500319074072478064120732846,
2.03954259792561744116669193558, 3.15987605895427165078636724539, 4.52016100892549047316912626960, 6.51539851266405452569228860040, 7.18437291149639858644195074932, 8.608723514576209367941208707134, 9.547148113734412827076121675315, 10.48618113559989277544079735893, 11.18055426766215539498024103686, 11.94420156713270657040853268969