L(s) = 1 | + (0.628 − 0.777i)2-s + (−0.960 − 0.277i)3-s + (−0.209 − 0.977i)4-s + (−0.998 − 0.0468i)5-s + (−0.819 + 0.572i)6-s + (−0.344 + 0.938i)7-s + (−0.892 − 0.451i)8-s + (0.845 + 0.533i)9-s + (−0.664 + 0.747i)10-s + (−0.762 + 0.646i)11-s + (−0.0702 + 0.997i)12-s + (0.664 − 0.747i)13-s + (0.513 + 0.858i)14-s + (0.946 + 0.322i)15-s + (−0.912 + 0.409i)16-s + (0.762 + 0.646i)17-s + ⋯ |
L(s) = 1 | + (0.628 − 0.777i)2-s + (−0.960 − 0.277i)3-s + (−0.209 − 0.977i)4-s + (−0.998 − 0.0468i)5-s + (−0.819 + 0.572i)6-s + (−0.344 + 0.938i)7-s + (−0.892 − 0.451i)8-s + (0.845 + 0.533i)9-s + (−0.664 + 0.747i)10-s + (−0.762 + 0.646i)11-s + (−0.0702 + 0.997i)12-s + (0.664 − 0.747i)13-s + (0.513 + 0.858i)14-s + (0.946 + 0.322i)15-s + (−0.912 + 0.409i)16-s + (0.762 + 0.646i)17-s + ⋯ |
Λ(s)=(=(269s/2ΓR(s)L(s)(0.883+0.469i)Λ(1−s)
Λ(s)=(=(269s/2ΓR(s)L(s)(0.883+0.469i)Λ(1−s)
Degree: |
1 |
Conductor: |
269
|
Sign: |
0.883+0.469i
|
Analytic conductor: |
1.24923 |
Root analytic conductor: |
1.24923 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ269(11,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 269, (0: ), 0.883+0.469i)
|
Particular Values
L(21) |
≈ |
0.5917353001+0.1474411103i |
L(21) |
≈ |
0.5917353001+0.1474411103i |
L(1) |
≈ |
0.7221616414−0.2107511453i |
L(1) |
≈ |
0.7221616414−0.2107511453i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 269 | 1 |
good | 2 | 1+(0.628−0.777i)T |
| 3 | 1+(−0.960−0.277i)T |
| 5 | 1+(−0.998−0.0468i)T |
| 7 | 1+(−0.344+0.938i)T |
| 11 | 1+(−0.762+0.646i)T |
| 13 | 1+(0.664−0.747i)T |
| 17 | 1+(0.762+0.646i)T |
| 19 | 1+(−0.731+0.681i)T |
| 23 | 1+(0.960+0.277i)T |
| 29 | 1+(−0.995+0.0936i)T |
| 31 | 1+(−0.163+0.986i)T |
| 37 | 1+(0.930+0.366i)T |
| 41 | 1+(−0.628−0.777i)T |
| 43 | 1+(0.995−0.0936i)T |
| 47 | 1+(−0.972+0.232i)T |
| 53 | 1+(0.255+0.966i)T |
| 59 | 1+(0.209+0.977i)T |
| 61 | 1+(−0.300+0.953i)T |
| 67 | 1+(−0.209+0.977i)T |
| 71 | 1+(−0.513+0.858i)T |
| 73 | 1+(−0.472+0.881i)T |
| 79 | 1+(−0.0234−0.999i)T |
| 83 | 1+(−0.430−0.902i)T |
| 89 | 1+(−0.388+0.921i)T |
| 97 | 1+(−0.300−0.953i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−25.865997546077106703665601048865, −24.29931589238335879969563298212, −23.6431548251430206574345540079, −23.18519175140036943746619523398, −22.45506602329170241757670130368, −21.28485648006480014873886933829, −20.609795997560216156799203533302, −19.04947275085906735477263904247, −18.21185868368661821656785478399, −16.77047892529764141564496494382, −16.523563379582438951094685809979, −15.668477470998114007180855847986, −14.72943126658358164772430089678, −13.39581337550899251774462675711, −12.72623143539142640896356455997, −11.39699403694976584522940168867, −10.9933251613322062327156815766, −9.390418086612542025304492154834, −7.99846466291791974509281955453, −7.103621502349649841250060295418, −6.27037369103405391857294374590, −5.01256056372821491226486220086, −4.15019893426096218359706931062, −3.26007537682417902578600828902, −0.41435158753084940280834346761,
1.38009791406238039834861281535, 2.8982669117575945131453067896, 4.09448217837860172349418908154, 5.28009235565798133284702637634, 5.96426637047490059151645270676, 7.32896202283565940351280533994, 8.62644325040036949724668370231, 10.14324121603783188017756489617, 10.87988598283608782353528677442, 11.83762202737064455724730573809, 12.66772708389774066216937858708, 12.9915301459416645573978216365, 14.876467144765219210516973536800, 15.445791409718906839602370902825, 16.40297121405867948256993295086, 17.872447295135556974571451275327, 18.758008150093020526820065199395, 19.212181150260120607559722305863, 20.52255356491239583197580462622, 21.36127565345758325289150798202, 22.37569077837817456983521667485, 23.19250732367934181379402062070, 23.46623741579782383954826442607, 24.61221378804259818271896750921, 25.65640076149534603380036386846