L(s) = 1 | + (−0.913 − 0.406i)2-s + (0.669 + 0.743i)4-s + (−0.913 + 0.406i)5-s + (−0.309 − 0.951i)8-s + 10-s + (−0.104 − 0.994i)13-s + (−0.104 + 0.994i)16-s + (−0.809 − 0.587i)17-s + (0.309 + 0.951i)19-s + (−0.913 − 0.406i)20-s + (−0.5 + 0.866i)23-s + (0.669 − 0.743i)25-s + (−0.309 + 0.951i)26-s + (0.978 − 0.207i)29-s + (0.104 + 0.994i)31-s + (0.5 − 0.866i)32-s + ⋯ |
L(s) = 1 | + (−0.913 − 0.406i)2-s + (0.669 + 0.743i)4-s + (−0.913 + 0.406i)5-s + (−0.309 − 0.951i)8-s + 10-s + (−0.104 − 0.994i)13-s + (−0.104 + 0.994i)16-s + (−0.809 − 0.587i)17-s + (0.309 + 0.951i)19-s + (−0.913 − 0.406i)20-s + (−0.5 + 0.866i)23-s + (0.669 − 0.743i)25-s + (−0.309 + 0.951i)26-s + (0.978 − 0.207i)29-s + (0.104 + 0.994i)31-s + (0.5 − 0.866i)32-s + ⋯ |
Λ(s)=(=(693s/2ΓR(s)L(s)(−0.710+0.703i)Λ(1−s)
Λ(s)=(=(693s/2ΓR(s)L(s)(−0.710+0.703i)Λ(1−s)
Degree: |
1 |
Conductor: |
693
= 32⋅7⋅11
|
Sign: |
−0.710+0.703i
|
Analytic conductor: |
3.21827 |
Root analytic conductor: |
3.21827 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ693(160,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 693, (0: ), −0.710+0.703i)
|
Particular Values
L(21) |
≈ |
0.07263654715+0.1764903446i |
L(21) |
≈ |
0.07263654715+0.1764903446i |
L(1) |
≈ |
0.4961464675+0.01264106803i |
L(1) |
≈ |
0.4961464675+0.01264106803i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 7 | 1 |
| 11 | 1 |
good | 2 | 1+(−0.913−0.406i)T |
| 5 | 1+(−0.913+0.406i)T |
| 13 | 1+(−0.104−0.994i)T |
| 17 | 1+(−0.809−0.587i)T |
| 19 | 1+(0.309+0.951i)T |
| 23 | 1+(−0.5+0.866i)T |
| 29 | 1+(0.978−0.207i)T |
| 31 | 1+(0.104+0.994i)T |
| 37 | 1+(0.309−0.951i)T |
| 41 | 1+(−0.978−0.207i)T |
| 43 | 1+(0.5+0.866i)T |
| 47 | 1+(−0.669+0.743i)T |
| 53 | 1+(−0.809+0.587i)T |
| 59 | 1+(−0.669−0.743i)T |
| 61 | 1+(−0.104+0.994i)T |
| 67 | 1+(−0.5+0.866i)T |
| 71 | 1+(−0.809−0.587i)T |
| 73 | 1+(0.309−0.951i)T |
| 79 | 1+(−0.913−0.406i)T |
| 83 | 1+(−0.104+0.994i)T |
| 89 | 1−T |
| 97 | 1+(−0.913−0.406i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−22.47041131185394460059580454641, −21.454549725895908717098461763015, −20.3108887709620673704005499476, −19.89339577454795793567295156623, −19.036142330915556495722726361954, −18.396136416422338040557923355853, −17.31575019706803794160441407175, −16.68740308053356110338915808820, −15.837668188839149928269402579129, −15.28272380057725044370363215985, −14.35437338051014046288775105987, −13.26814729240331978378447196365, −12.02686071941840719834004529263, −11.45171551409079378228003838551, −10.567129105240827089875000576621, −9.50145753303096013016774856899, −8.6675537268534599894783760672, −8.08656777034248682090501348466, −6.98868186550718857204159166769, −6.39886788797099523859562215884, −4.9945830848400197697689379265, −4.19563031434456584609456439302, −2.71347573976907060904962621487, −1.5214200397571558926622328164, −0.1335631220189887012646808864,
1.310503628401838554617690094075, 2.725325252161988653206091678327, 3.4111917185048489409157310769, 4.51745191537637068376426544655, 6.00093690800126625818113631280, 7.08071370738723328290128409566, 7.79964178280618159737870130437, 8.47293721690279363755069417919, 9.57816702595631099491738197836, 10.43942165746833565665480691376, 11.15645361743108218098520766095, 11.99601801644689280994858801078, 12.64070186978733014245313322097, 13.86494576801166500994931255374, 15.01637604307330859839974658884, 15.83328104596840029497031839450, 16.28482312778959586987141836548, 17.649313348860407548519770351053, 17.994274152456054276037952276264, 18.99653310261149465078872140106, 19.71568995931997264587447459786, 20.23238122111550502269636135688, 21.16726840988187066914687390570, 22.16704719232926757372740300024, 22.84098419897869437084034581933