L(s) = 1 | + (−1.85 − 0.673i)2-s + (1.43 + 1.20i)4-s + (−0.642 + 3.64i)5-s + (−1.79 + 1.50i)7-s + (0.118 + 0.205i)8-s + (3.64 − 6.31i)10-s + (0.378 + 2.14i)11-s + (4.43 − 1.61i)13-s + (4.34 − 1.58i)14-s + (−0.733 − 4.16i)16-s + (−1.46 + 2.54i)17-s + (3.11 + 5.39i)19-s + (−5.32 + 4.47i)20-s + (0.745 − 4.22i)22-s + (−0.397 − 0.333i)23-s + ⋯ |
L(s) = 1 | + (−1.30 − 0.476i)2-s + (0.719 + 0.604i)4-s + (−0.287 + 1.63i)5-s + (−0.679 + 0.570i)7-s + (0.0419 + 0.0727i)8-s + (1.15 − 1.99i)10-s + (0.114 + 0.646i)11-s + (1.22 − 0.447i)13-s + (1.16 − 0.422i)14-s + (−0.183 − 1.04i)16-s + (−0.355 + 0.616i)17-s + (0.714 + 1.23i)19-s + (−1.19 + 0.999i)20-s + (0.158 − 0.900i)22-s + (−0.0829 − 0.0695i)23-s + ⋯ |
Λ(s)=(=(729s/2ΓC(s)L(s)(−0.802−0.597i)Λ(2−s)
Λ(s)=(=(729s/2ΓC(s+1/2)L(s)(−0.802−0.597i)Λ(1−s)
Degree: |
2 |
Conductor: |
729
= 36
|
Sign: |
−0.802−0.597i
|
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), −0.802−0.597i)
|
Particular Values
L(1) |
≈ |
0.139280+0.420323i |
L(21) |
≈ |
0.139280+0.420323i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
good | 2 | 1+(1.85+0.673i)T+(1.53+1.28i)T2 |
| 5 | 1+(0.642−3.64i)T+(−4.69−1.71i)T2 |
| 7 | 1+(1.79−1.50i)T+(1.21−6.89i)T2 |
| 11 | 1+(−0.378−2.14i)T+(−10.3+3.76i)T2 |
| 13 | 1+(−4.43+1.61i)T+(9.95−8.35i)T2 |
| 17 | 1+(1.46−2.54i)T+(−8.5−14.7i)T2 |
| 19 | 1+(−3.11−5.39i)T+(−9.5+16.4i)T2 |
| 23 | 1+(0.397+0.333i)T+(3.99+22.6i)T2 |
| 29 | 1+(3.28+1.19i)T+(22.2+18.6i)T2 |
| 31 | 1+(3.29+2.76i)T+(5.38+30.5i)T2 |
| 37 | 1+(1.20−2.08i)T+(−18.5−32.0i)T2 |
| 41 | 1+(−2.34+0.854i)T+(31.4−26.3i)T2 |
| 43 | 1+(−0.184−1.04i)T+(−40.4+14.7i)T2 |
| 47 | 1+(−0.181+0.152i)T+(8.16−46.2i)T2 |
| 53 | 1+4.66T+53T2 |
| 59 | 1+(−2.31+13.1i)T+(−55.4−20.1i)T2 |
| 61 | 1+(2.81−2.36i)T+(10.5−60.0i)T2 |
| 67 | 1+(13.4−4.89i)T+(51.3−43.0i)T2 |
| 71 | 1+(−0.601+1.04i)T+(−35.5−61.4i)T2 |
| 73 | 1+(−2.34−4.05i)T+(−36.5+63.2i)T2 |
| 79 | 1+(−12.0−4.37i)T+(60.5+50.7i)T2 |
| 83 | 1+(10.6+3.86i)T+(63.5+53.3i)T2 |
| 89 | 1+(−0.349−0.605i)T+(−44.5+77.0i)T2 |
| 97 | 1+(1.23+6.97i)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
−10.61701879536203007169243421526, −9.935048185876011715841622838099, −9.249706403702547839353833114583, −8.172849375139151043759602702134, −7.53248846063986408914228868227, −6.53623921474587373849393076359, −5.75005327870241179247453388393, −3.80186050372204447593869314525, −2.93225009246355363228501668370, −1.78278160196351368155497694082,
0.39157925542589686518637285951, 1.32651575353131340359931315930, 3.56394644721511209215636070520, 4.56688932750607381378120965244, 5.77953013232616860803247058955, 6.84145834298669143783990223786, 7.62284217410166765040013337741, 8.643451151275889276944172588992, 9.024193411210011766179354412828, 9.564310965217071656036677430047