L(s) = 1 | + (0.300 − 1.70i)2-s + (−0.939 − 0.342i)4-s + (2.65 − 2.22i)5-s + (0.939 − 0.342i)7-s + (0.866 − 1.50i)8-s + (−2.99 − 5.19i)10-s + (−2.65 − 2.22i)11-s + (0.868 + 4.92i)13-s + (−0.300 − 1.70i)14-s + (−3.83 − 3.21i)16-s + (0.5 − 0.866i)19-s + (−3.25 + 1.18i)20-s + (−4.59 + 3.85i)22-s + (6.51 + 2.36i)23-s + (1.21 − 6.89i)25-s + 8.66·26-s + ⋯ |
L(s) = 1 | + (0.212 − 1.20i)2-s + (−0.469 − 0.171i)4-s + (1.18 − 0.995i)5-s + (0.355 − 0.129i)7-s + (0.306 − 0.530i)8-s + (−0.948 − 1.64i)10-s + (−0.800 − 0.671i)11-s + (0.240 + 1.36i)13-s + (−0.0803 − 0.455i)14-s + (−0.957 − 0.803i)16-s + (0.114 − 0.198i)19-s + (−0.727 + 0.264i)20-s + (−0.979 + 0.822i)22-s + (1.35 + 0.494i)23-s + (0.243 − 1.37i)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(649,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 729, ( :1/2), −0.686+0.727i)
|
Particular Values
L(1) |
≈ |
0.883081−2.04721i |
L(21) |
≈ |
0.883081−2.04721i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
good | 2 | 1+(−0.300+1.70i)T+(−1.87−0.684i)T2 |
| 5 | 1+(−2.65+2.22i)T+(0.868−4.92i)T2 |
| 7 | 1+(−0.939+0.342i)T+(5.36−4.49i)T2 |
| 11 | 1+(2.65+2.22i)T+(1.91+10.8i)T2 |
| 13 | 1+(−0.868−4.92i)T+(−12.2+4.44i)T2 |
| 17 | 1+(−8.5+14.7i)T2 |
| 19 | 1+(−0.5+0.866i)T+(−9.5−16.4i)T2 |
| 23 | 1+(−6.51−2.36i)T+(17.6+14.7i)T2 |
| 29 | 1+(0.601−3.41i)T+(−27.2−9.91i)T2 |
| 31 | 1+(4.69+1.71i)T+(23.7+19.9i)T2 |
| 37 | 1+(−0.5−0.866i)T+(−18.5+32.0i)T2 |
| 41 | 1+(−0.601−3.41i)T+(−38.5+14.0i)T2 |
| 43 | 1+(0.766+0.642i)T+(7.46+42.3i)T2 |
| 47 | 1+(−3.25+1.18i)T+(36.0−30.2i)T2 |
| 53 | 1+10.3T+53T2 |
| 59 | 1+(−2.65+2.22i)T+(10.2−58.1i)T2 |
| 61 | 1+(1.87−0.684i)T+(46.7−39.2i)T2 |
| 67 | 1+(−1.38−7.87i)T+(−62.9+22.9i)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+(0.173−0.984i)T+(−74.2−27.0i)T2 |
| 83 | 1+(−1.20+6.82i)T+(−77.9−28.3i)T2 |
| 89 | 1+(5.19−9i)T+(−44.5−77.0i)T2 |
| 97 | 1+(−13.0−10.9i)T+(16.8+95.5i)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
−10.18700722054856216514550724547, −9.277425558064261322990905750510, −8.876465974020775582839104507821, −7.51873973536084641363063118434, −6.37868333126960020429288477545, −5.25060921636549554577470820489, −4.55973426849479449646016263257, −3.24614602853304289953305633151, −2.04163807764496035973278188877, −1.19475357274974093344576100748,
2.02264235311953138203832965823, 3.03874877494723370325247381901, 4.87262714758576893241418265315, 5.55511068883109287679616244150, 6.25121445900939175952682249904, 7.17643415295584154777043335855, 7.77910628578581401433833208507, 8.797693011515744597603713072820, 9.946108894841672447181167750410, 10.61249588407867013252312480020