L(s) = 1 | + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.104 + 0.994i)5-s + (−0.309 − 0.951i)8-s + (0.5 + 0.866i)10-s + (0.104 + 0.994i)13-s + (−0.809 − 0.587i)16-s + (−0.104 + 0.994i)17-s + (0.978 + 0.207i)19-s + (0.913 + 0.406i)20-s + (0.5 + 0.866i)23-s + (−0.978 − 0.207i)25-s + (0.669 + 0.743i)26-s + (0.978 − 0.207i)29-s + (0.809 − 0.587i)31-s − 32-s + ⋯ |
L(s) = 1 | + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.104 + 0.994i)5-s + (−0.309 − 0.951i)8-s + (0.5 + 0.866i)10-s + (0.104 + 0.994i)13-s + (−0.809 − 0.587i)16-s + (−0.104 + 0.994i)17-s + (0.978 + 0.207i)19-s + (0.913 + 0.406i)20-s + (0.5 + 0.866i)23-s + (−0.978 − 0.207i)25-s + (0.669 + 0.743i)26-s + (0.978 − 0.207i)29-s + (0.809 − 0.587i)31-s − 32-s + ⋯ |
Λ(s)=(=(693s/2ΓR(s)L(s)(0.999+0.0111i)Λ(1−s)
Λ(s)=(=(693s/2ΓR(s)L(s)(0.999+0.0111i)Λ(1−s)
Degree: |
1 |
Conductor: |
693
= 32⋅7⋅11
|
Sign: |
0.999+0.0111i
|
Analytic conductor: |
3.21827 |
Root analytic conductor: |
3.21827 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ693(38,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 693, (0: ), 0.999+0.0111i)
|
Particular Values
L(21) |
≈ |
2.213881660+0.01237308261i |
L(21) |
≈ |
2.213881660+0.01237308261i |
L(1) |
≈ |
1.618373319−0.2038402562i |
L(1) |
≈ |
1.618373319−0.2038402562i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 7 | 1 |
| 11 | 1 |
good | 2 | 1+(0.809−0.587i)T |
| 5 | 1+(−0.104+0.994i)T |
| 13 | 1+(0.104+0.994i)T |
| 17 | 1+(−0.104+0.994i)T |
| 19 | 1+(0.978+0.207i)T |
| 23 | 1+(0.5+0.866i)T |
| 29 | 1+(0.978−0.207i)T |
| 31 | 1+(0.809−0.587i)T |
| 37 | 1+(0.669+0.743i)T |
| 41 | 1+(−0.978−0.207i)T |
| 43 | 1+(−0.5−0.866i)T |
| 47 | 1+(0.309+0.951i)T |
| 53 | 1+(−0.913−0.406i)T |
| 59 | 1+(0.309−0.951i)T |
| 61 | 1+(0.809+0.587i)T |
| 67 | 1+T |
| 71 | 1+(0.809+0.587i)T |
| 73 | 1+(0.978−0.207i)T |
| 79 | 1+(−0.809+0.587i)T |
| 83 | 1+(−0.104+0.994i)T |
| 89 | 1+(−0.5+0.866i)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
−22.908857514787956245096668635411, −21.942015187597164814853685290661, −21.10484546932752974678234301799, −20.31755155391102220399966061829, −19.87573486247376755282297904132, −18.32589451253294603362136137441, −17.587315844773553370765030557957, −16.71185886426569663524082414656, −15.966349298681529609133972757072, −15.448194954698315154074779993047, −14.3259079919386297692609343951, −13.54105014842531231162724828352, −12.80844814299202206428276372516, −12.08394628937959625017773344564, −11.28516253957277109146090261520, −9.97471066384843855063814932736, −8.84358689083618125626402938734, −8.145950259343070266876452602468, −7.25424955174794559743089202334, −6.22766763084551298579239174760, −5.10722852761659233975230206421, −4.78736901260554166463130038158, −3.48447728329455576560395242052, −2.57781176463634300448845659211, −0.90542689687867168897558352780,
1.373332346507290039085328353473, 2.43570526421544571971300286684, 3.39423063612475982125602097569, 4.16114376899510061188800282013, 5.30708956733430886676836588758, 6.35678011852847890792407422291, 6.92953896198014103717194473571, 8.16360835011232837878126546767, 9.58280423426148386923772287512, 10.162022126837895394322211623265, 11.263225296393655920906254398737, 11.62384992652217282254950285417, 12.679512876261690756247570033928, 13.786538874296533483579858781712, 14.13346127876365704661949371374, 15.21348941595111058874368549963, 15.670751439703252286965810495437, 16.93885217723983471926630639389, 18.016899945518633732483544549205, 18.96052249160934014101249630173, 19.30031222209300461458770950223, 20.34485206112707674288525755824, 21.24496983580773063803100513419, 21.92339965216829295883727346650, 22.49129574233981375857861659314