L(s) = 1 | − i·2-s − 4-s + (−0.866 − 0.5i)5-s + i·8-s + (−0.5 + 0.866i)10-s + (−0.866 − 0.5i)11-s + 16-s − 17-s + (−0.866 + 0.5i)19-s + (0.866 + 0.5i)20-s + (−0.5 + 0.866i)22-s + 23-s + (0.5 + 0.866i)25-s + (0.5 + 0.866i)29-s + (−0.866 + 0.5i)31-s − i·32-s + ⋯ |
L(s) = 1 | − i·2-s − 4-s + (−0.866 − 0.5i)5-s + i·8-s + (−0.5 + 0.866i)10-s + (−0.866 − 0.5i)11-s + 16-s − 17-s + (−0.866 + 0.5i)19-s + (0.866 + 0.5i)20-s + (−0.5 + 0.866i)22-s + 23-s + (0.5 + 0.866i)25-s + (0.5 + 0.866i)29-s + (−0.866 + 0.5i)31-s − i·32-s + ⋯ |
Λ(s)=(=(273s/2ΓR(s+1)L(s)(0.931−0.362i)Λ(1−s)
Λ(s)=(=(273s/2ΓR(s+1)L(s)(0.931−0.362i)Λ(1−s)
Degree: |
1 |
Conductor: |
273
= 3⋅7⋅13
|
Sign: |
0.931−0.362i
|
Analytic conductor: |
29.3379 |
Root analytic conductor: |
29.3379 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ273(59,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 273, (1: ), 0.931−0.362i)
|
Particular Values
L(21) |
≈ |
0.7754945332−0.1455506885i |
L(21) |
≈ |
0.7754945332−0.1455506885i |
L(1) |
≈ |
0.6293141631−0.3395415936i |
L(1) |
≈ |
0.6293141631−0.3395415936i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 7 | 1 |
| 13 | 1 |
good | 2 | 1−iT |
| 5 | 1+(−0.866−0.5i)T |
| 11 | 1+(−0.866−0.5i)T |
| 17 | 1−T |
| 19 | 1+(−0.866+0.5i)T |
| 23 | 1+T |
| 29 | 1+(0.5+0.866i)T |
| 31 | 1+(−0.866+0.5i)T |
| 37 | 1−iT |
| 41 | 1+(0.866−0.5i)T |
| 43 | 1+(0.5−0.866i)T |
| 47 | 1+(0.866+0.5i)T |
| 53 | 1+(0.5+0.866i)T |
| 59 | 1−iT |
| 61 | 1+(0.5+0.866i)T |
| 67 | 1+(−0.866−0.5i)T |
| 71 | 1+(0.866+0.5i)T |
| 73 | 1+(0.866−0.5i)T |
| 79 | 1+(−0.5+0.866i)T |
| 83 | 1+iT |
| 89 | 1−iT |
| 97 | 1+(0.866+0.5i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−25.68568425064696055877359976154, −24.54325639381457856169454267229, −23.68600852791192979728154353151, −23.055240660359563643180566992817, −22.25619471069483904423763959447, −21.164173411593377316918643003593, −19.82553375035822121421586465758, −18.93894467561208807643973623217, −18.10027637359264069747866770837, −17.21449384609231705661571211004, −16.07598376304677219612008868246, −15.285612099493965970686758303024, −14.79748909179351866969788547560, −13.4348198513080522539962599634, −12.69339747401635579870172525180, −11.305429604706831573954771130620, −10.305103355705523333297509034504, −9.023903162913477914738732448762, −8.04215313023448854280985628904, −7.196574529357040579096299366361, −6.330185539836268578123131408824, −4.904188171468875253241031879119, −4.09138862500920968463255462747, −2.63748572402811903098509335979, −0.365099683096231204695746554458,
0.78428597902106022152610446420, 2.32639209638966839841679347823, 3.55581777194456291415610084679, 4.530047138634501930478734877993, 5.54790997110994000143148281867, 7.32464668921984999888269266102, 8.50132219481796349355355669913, 9.07530978276759813634186775534, 10.686532477386080624092741444703, 11.054788455178144215005846880010, 12.41318566184063163877657660088, 12.86707049135461707135335171365, 14.020752844477107491210321204146, 15.19941078792704212991869011808, 16.21068692899313206566154192066, 17.28437072399372759195613642897, 18.37737730567241552967052686960, 19.19777747419815073481538049103, 19.939980875611256660573159557620, 20.838117105862193899371051878985, 21.56888599153690307687906585963, 22.70645752861919287065080380088, 23.50856513513089402393185438341, 24.18227123514275738024196101254, 25.57277891711903410278162699224