L(s) = 1 | + (0.978 − 0.207i)2-s + (0.913 − 0.406i)4-s + (−0.309 − 0.951i)5-s + (0.809 − 0.587i)8-s + (−0.5 − 0.866i)10-s + (−0.978 + 0.207i)13-s + (0.669 − 0.743i)16-s + (0.669 − 0.743i)17-s + (−0.104 − 0.994i)19-s + (−0.669 − 0.743i)20-s + 23-s + (−0.809 + 0.587i)25-s + (−0.913 + 0.406i)26-s + (−0.913 + 0.406i)29-s + (−0.669 − 0.743i)31-s + (0.5 − 0.866i)32-s + ⋯ |
L(s) = 1 | + (0.978 − 0.207i)2-s + (0.913 − 0.406i)4-s + (−0.309 − 0.951i)5-s + (0.809 − 0.587i)8-s + (−0.5 − 0.866i)10-s + (−0.978 + 0.207i)13-s + (0.669 − 0.743i)16-s + (0.669 − 0.743i)17-s + (−0.104 − 0.994i)19-s + (−0.669 − 0.743i)20-s + 23-s + (−0.809 + 0.587i)25-s + (−0.913 + 0.406i)26-s + (−0.913 + 0.406i)29-s + (−0.669 − 0.743i)31-s + (0.5 − 0.866i)32-s + ⋯ |
Λ(s)=(=(693s/2ΓR(s)L(s)(−0.152−0.988i)Λ(1−s)
Λ(s)=(=(693s/2ΓR(s)L(s)(−0.152−0.988i)Λ(1−s)
Degree: |
1 |
Conductor: |
693
= 32⋅7⋅11
|
Sign: |
−0.152−0.988i
|
Analytic conductor: |
3.21827 |
Root analytic conductor: |
3.21827 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ693(283,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 693, (0: ), −0.152−0.988i)
|
Particular Values
L(21) |
≈ |
1.547891572−1.804481886i |
L(21) |
≈ |
1.547891572−1.804481886i |
L(1) |
≈ |
1.594054766−0.7522049403i |
L(1) |
≈ |
1.594054766−0.7522049403i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 7 | 1 |
| 11 | 1 |
good | 2 | 1+(0.978−0.207i)T |
| 5 | 1+(−0.309−0.951i)T |
| 13 | 1+(−0.978+0.207i)T |
| 17 | 1+(0.669−0.743i)T |
| 19 | 1+(−0.104−0.994i)T |
| 23 | 1+T |
| 29 | 1+(−0.913+0.406i)T |
| 31 | 1+(−0.669−0.743i)T |
| 37 | 1+(0.913−0.406i)T |
| 41 | 1+(0.913+0.406i)T |
| 43 | 1+(0.5−0.866i)T |
| 47 | 1+(−0.913−0.406i)T |
| 53 | 1+(−0.978+0.207i)T |
| 59 | 1+(−0.913+0.406i)T |
| 61 | 1+(0.669−0.743i)T |
| 67 | 1+(−0.5+0.866i)T |
| 71 | 1+(0.309+0.951i)T |
| 73 | 1+(−0.104+0.994i)T |
| 79 | 1+(0.978−0.207i)T |
| 83 | 1+(−0.978−0.207i)T |
| 89 | 1+(0.5−0.866i)T |
| 97 | 1+(−0.669−0.743i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−22.84800132795184834088940435418, −22.251537020543468461087653124662, −21.43514263364700237376699608285, −20.71803909147991661828581277556, −19.576751107416053844305641795651, −19.10206400084155241744770633437, −17.934963324657026676064143318140, −16.94803043297351395882985351737, −16.24530549812161194555763757091, −15.03769139836511078651288122903, −14.78731611336676601403705505217, −14.01930654127644309948483254761, −12.83458166375642724529215816444, −12.27613985441691184787391203273, −11.23579003756696454162397374900, −10.60720498241078842451152479487, −9.590753942536407216611938845676, −7.95713328658588371917292163310, −7.51158688754498602189220186075, −6.47856186806685828179713154402, −5.71103665365870415397457369965, −4.62631268611984144270663378689, −3.591518170217294438317501660738, −2.88780130808336526963002414543, −1.76345300685500697720712851130,
0.81686369999560757331786320023, 2.117432662222391978941778409524, 3.159268309334743583552231850701, 4.30404603424474896218714610614, 4.98912885668614671905238722872, 5.72916237316519667145122709289, 7.08574452575894774287267454478, 7.66197030832443180527951287915, 9.08963925451947467850460047783, 9.76629758544463119231464922616, 11.12216542951319609623551144511, 11.65392185537559527089408027431, 12.713320941990163856813950786796, 13.03083714151492083765083008857, 14.18287717917786737762653756051, 14.922555449415257804211545903704, 15.783087269209314522640668507863, 16.59101156222879194111761650255, 17.20963551200702760130232547786, 18.63913495183862121152051238748, 19.5158498624786305020594228814, 20.15421444988383711992217294761, 20.87244318978128938354851536822, 21.63759631420473239513123282193, 22.43071821989021306047943365493