L(s) = 1 | + (0.365 − 0.930i)2-s + (−0.733 − 0.680i)4-s + (0.0747 − 0.997i)5-s + (−0.900 + 0.433i)8-s + (−0.900 − 0.433i)10-s + (0.365 − 0.930i)11-s + (−0.988 + 0.149i)13-s + (0.0747 + 0.997i)16-s + (−0.222 − 0.974i)17-s + 19-s + (−0.733 + 0.680i)20-s + (−0.733 − 0.680i)22-s + (−0.733 − 0.680i)23-s + (−0.988 − 0.149i)25-s + (−0.222 + 0.974i)26-s + ⋯ |
L(s) = 1 | + (0.365 − 0.930i)2-s + (−0.733 − 0.680i)4-s + (0.0747 − 0.997i)5-s + (−0.900 + 0.433i)8-s + (−0.900 − 0.433i)10-s + (0.365 − 0.930i)11-s + (−0.988 + 0.149i)13-s + (0.0747 + 0.997i)16-s + (−0.222 − 0.974i)17-s + 19-s + (−0.733 + 0.680i)20-s + (−0.733 − 0.680i)22-s + (−0.733 − 0.680i)23-s + (−0.988 − 0.149i)25-s + (−0.222 + 0.974i)26-s + ⋯ |
Λ(s)=(=(441s/2ΓR(s)L(s)(−0.944+0.328i)Λ(1−s)
Λ(s)=(=(441s/2ΓR(s)L(s)(−0.944+0.328i)Λ(1−s)
Degree: |
1 |
Conductor: |
441
= 32⋅72
|
Sign: |
−0.944+0.328i
|
Analytic conductor: |
2.04799 |
Root analytic conductor: |
2.04799 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ441(421,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 441, (0: ), −0.944+0.328i)
|
Particular Values
L(21) |
≈ |
−0.1746816274−1.033675420i |
L(21) |
≈ |
−0.1746816274−1.033675420i |
L(1) |
≈ |
0.6469327873−0.7892277685i |
L(1) |
≈ |
0.6469327873−0.7892277685i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 7 | 1 |
good | 2 | 1+(0.365−0.930i)T |
| 5 | 1+(0.0747−0.997i)T |
| 11 | 1+(0.365−0.930i)T |
| 13 | 1+(−0.988+0.149i)T |
| 17 | 1+(−0.222−0.974i)T |
| 19 | 1+T |
| 23 | 1+(−0.733−0.680i)T |
| 29 | 1+(−0.733+0.680i)T |
| 31 | 1+(−0.5+0.866i)T |
| 37 | 1+(−0.222−0.974i)T |
| 41 | 1+(0.0747−0.997i)T |
| 43 | 1+(0.0747+0.997i)T |
| 47 | 1+(0.365−0.930i)T |
| 53 | 1+(−0.222+0.974i)T |
| 59 | 1+(0.826−0.563i)T |
| 61 | 1+(−0.733+0.680i)T |
| 67 | 1+(−0.5+0.866i)T |
| 71 | 1+(−0.222+0.974i)T |
| 73 | 1+(0.623−0.781i)T |
| 79 | 1+(−0.5−0.866i)T |
| 83 | 1+(−0.988−0.149i)T |
| 89 | 1+(0.623−0.781i)T |
| 97 | 1+(−0.5−0.866i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−24.43149330654233151595052601727, −23.794723831742040784001543030649, −22.58545159626929412693549666044, −22.36538055880878941395995149387, −21.50803229074946821149432848111, −20.25659467336353323556116321937, −19.22797244439228884998893799940, −18.23100919293364002094343489075, −17.51820527063934659762559987927, −16.840655704769162575581039677348, −15.4989633473220251060131460656, −15.02452735622854038616954613943, −14.26762619486831315835534723222, −13.367040226824927157442747356515, −12.32464665141372619949809899954, −11.4619110556134249723576943573, −10.00415798875657514998778362966, −9.457959391995327752004316590555, −7.91788522631816849719678060531, −7.34631774815741734154093036173, −6.42455242543440210775413199361, −5.507281204133779648746489000265, −4.315992121815807151158520717919, −3.37250389154607994679385152155, −2.09262158218201882934393047399,
0.516458658469297684226994088174, 1.75687275722689466011271447268, 2.975166476944110360766656861869, 4.106899739773066911446711857649, 5.08006079694338624635519419224, 5.786908517682648705152559594090, 7.333434069495798061566187507915, 8.73090693762941341508725189072, 9.262820503382432602824047450506, 10.247138726268978995877411334125, 11.41469396041919088869231821379, 12.09130616653724407823508261360, 12.86738721082742133615784945845, 13.890393292393691191235216593200, 14.4104174230259993653886452653, 15.85978545989790163419480348396, 16.61133971045696239159117903378, 17.70570386510855153748795449578, 18.55331063996036425990639689936, 19.62984726304488061336921286208, 20.13692107462709363348341495740, 20.95845468564544666933679768586, 21.87784170760376334836514801303, 22.42681282856406370605953587987, 23.59812851180920152094318151024