L(s) = 1 | + (−1.62 − 0.592i)2-s + (0.766 + 0.642i)4-s + (0.601 − 3.41i)5-s + (−0.766 + 0.642i)7-s + (0.866 + 1.50i)8-s + (−3 + 5.19i)10-s + (−0.601 − 3.41i)11-s + (−4.69 + 1.71i)13-s + (1.62 − 0.592i)14-s + (−0.868 − 4.92i)16-s + (0.5 + 0.866i)19-s + (2.65 − 2.22i)20-s + (−1.04 + 5.90i)22-s + (−5.30 − 4.45i)23-s + (−6.57 − 2.39i)25-s + 8.66·26-s + ⋯ |
L(s) = 1 | + (−1.15 − 0.418i)2-s + (0.383 + 0.321i)4-s + (0.269 − 1.52i)5-s + (−0.289 + 0.242i)7-s + (0.306 + 0.530i)8-s + (−0.948 + 1.64i)10-s + (−0.181 − 1.02i)11-s + (−1.30 + 0.474i)13-s + (0.434 − 0.158i)14-s + (−0.217 − 1.23i)16-s + (0.114 + 0.198i)19-s + (0.593 − 0.497i)20-s + (−0.222 + 1.25i)22-s + (−1.10 − 0.928i)23-s + (−1.31 − 0.478i)25-s + 1.69·26-s + ⋯ |
Λ(s)=(=(729s/2ΓC(s)L(s)(−0.686−0.727i)Λ(2−s)
Λ(s)=(=(729s/2ΓC(s+1/2)L(s)(−0.686−0.727i)Λ(1−s)
Degree: |
2 |
Conductor: |
729
= 36
|
Sign: |
−0.686−0.727i
|
Analytic conductor: |
5.82109 |
Root analytic conductor: |
2.41269 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ729(568,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 729, ( :1/2), −0.686−0.727i)
|
Particular Values
L(1) |
≈ |
0.0849221+0.196871i |
L(21) |
≈ |
0.0849221+0.196871i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
good | 2 | 1+(1.62+0.592i)T+(1.53+1.28i)T2 |
| 5 | 1+(−0.601+3.41i)T+(−4.69−1.71i)T2 |
| 7 | 1+(0.766−0.642i)T+(1.21−6.89i)T2 |
| 11 | 1+(0.601+3.41i)T+(−10.3+3.76i)T2 |
| 13 | 1+(4.69−1.71i)T+(9.95−8.35i)T2 |
| 17 | 1+(−8.5−14.7i)T2 |
| 19 | 1+(−0.5−0.866i)T+(−9.5+16.4i)T2 |
| 23 | 1+(5.30+4.45i)T+(3.99+22.6i)T2 |
| 29 | 1+(−3.25−1.18i)T+(22.2+18.6i)T2 |
| 31 | 1+(−3.83−3.21i)T+(5.38+30.5i)T2 |
| 37 | 1+(−0.5+0.866i)T+(−18.5−32.0i)T2 |
| 41 | 1+(3.25−1.18i)T+(31.4−26.3i)T2 |
| 43 | 1+(0.173+0.984i)T+(−40.4+14.7i)T2 |
| 47 | 1+(2.65−2.22i)T+(8.16−46.2i)T2 |
| 53 | 1+10.3T+53T2 |
| 59 | 1+(−0.601+3.41i)T+(−55.4−20.1i)T2 |
| 61 | 1+(−1.53+1.28i)T+(10.5−60.0i)T2 |
| 67 | 1+(7.51−2.73i)T+(51.3−43.0i)T2 |
| 71 | 1+(5.19−9i)T+(−35.5−61.4i)T2 |
| 73 | 1+(1+1.73i)T+(−36.5+63.2i)T2 |
| 79 | 1+(−0.939−0.342i)T+(60.5+50.7i)T2 |
| 83 | 1+(6.51+2.36i)T+(63.5+53.3i)T2 |
| 89 | 1+(5.19+9i)T+(−44.5+77.0i)T2 |
| 97 | 1+(−2.95−16.7i)T+(−91.1+33.1i)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
−9.766320771936237590222635640635, −8.990777945500777974402238667564, −8.456655767041508436778629654314, −7.74107228751914331902975421069, −6.29961580013877011494238769413, −5.21657060901639375665413963958, −4.50257540807675856986974457865, −2.69807831960993803829691003113, −1.44927442163203156253429363928, −0.16261469320101544249087164867,
2.06873494140408445931418024130, 3.23168591085208729326122769740, 4.56536829063430762538325492335, 6.04615678039507940948291272271, 6.96484897228611408026422422170, 7.40534810118907594995579701909, 8.146393916877400854772921045049, 9.568081739921691600570863820913, 9.978426038081451576149715442264, 10.37734859859672889569519109745