L(s) = 1 | + (0.912 + 0.409i)2-s + (−0.430 + 0.902i)3-s + (0.664 + 0.747i)4-s + (0.982 − 0.186i)5-s + (−0.762 + 0.646i)6-s + (−0.163 + 0.986i)7-s + (0.300 + 0.953i)8-s + (−0.628 − 0.777i)9-s + (0.972 + 0.232i)10-s + (−0.946 + 0.322i)11-s + (−0.960 + 0.277i)12-s + (−0.972 − 0.232i)13-s + (−0.553 + 0.833i)14-s + (−0.255 + 0.966i)15-s + (−0.116 + 0.993i)16-s + (0.946 + 0.322i)17-s + ⋯ |
L(s) = 1 | + (0.912 + 0.409i)2-s + (−0.430 + 0.902i)3-s + (0.664 + 0.747i)4-s + (0.982 − 0.186i)5-s + (−0.762 + 0.646i)6-s + (−0.163 + 0.986i)7-s + (0.300 + 0.953i)8-s + (−0.628 − 0.777i)9-s + (0.972 + 0.232i)10-s + (−0.946 + 0.322i)11-s + (−0.960 + 0.277i)12-s + (−0.972 − 0.232i)13-s + (−0.553 + 0.833i)14-s + (−0.255 + 0.966i)15-s + (−0.116 + 0.993i)16-s + (0.946 + 0.322i)17-s + ⋯ |
Λ(s)=(=(269s/2ΓR(s)L(s)(−0.522+0.852i)Λ(1−s)
Λ(s)=(=(269s/2ΓR(s)L(s)(−0.522+0.852i)Λ(1−s)
Degree: |
1 |
Conductor: |
269
|
Sign: |
−0.522+0.852i
|
Analytic conductor: |
1.24923 |
Root analytic conductor: |
1.24923 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ269(138,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 269, (0: ), −0.522+0.852i)
|
Particular Values
L(21) |
≈ |
0.9499191151+1.695655494i |
L(21) |
≈ |
0.9499191151+1.695655494i |
L(1) |
≈ |
1.282115032+1.018826026i |
L(1) |
≈ |
1.282115032+1.018826026i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 269 | 1 |
good | 2 | 1+(0.912+0.409i)T |
| 3 | 1+(−0.430+0.902i)T |
| 5 | 1+(0.982−0.186i)T |
| 7 | 1+(−0.163+0.986i)T |
| 11 | 1+(−0.946+0.322i)T |
| 13 | 1+(−0.972−0.232i)T |
| 17 | 1+(0.946+0.322i)T |
| 19 | 1+(0.990−0.140i)T |
| 23 | 1+(0.430−0.902i)T |
| 29 | 1+(−0.930−0.366i)T |
| 31 | 1+(−0.792+0.610i)T |
| 37 | 1+(0.0702−0.997i)T |
| 41 | 1+(−0.912+0.409i)T |
| 43 | 1+(0.930+0.366i)T |
| 47 | 1+(0.591+0.806i)T |
| 53 | 1+(0.513+0.858i)T |
| 59 | 1+(−0.664−0.747i)T |
| 61 | 1+(0.344−0.938i)T |
| 67 | 1+(0.664−0.747i)T |
| 71 | 1+(0.553+0.833i)T |
| 73 | 1+(−0.388−0.921i)T |
| 79 | 1+(0.995+0.0936i)T |
| 83 | 1+(0.209−0.977i)T |
| 89 | 1+(−0.0234−0.999i)T |
| 97 | 1+(0.344+0.938i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−25.2495756184049732300963313775, −24.189753952478673426790156002013, −23.71394557109879248017865681955, −22.68117272855738798963450566393, −22.038118213317337429138236885024, −20.93445935744064103761298411802, −20.13962091520060665122937461795, −19.020518869618475822666779161885, −18.29686564598423006390766152386, −17.06462727963415160360068533570, −16.39147490678901095401506007635, −14.80111996479573654505245004564, −13.76621047840113660907920138227, −13.44358644444252254669288689089, −12.46685932789550095497960855629, −11.408686869409408282711689367374, −10.4515903332640262948099314760, −9.652527497294128327403400947, −7.51341597059192850989599569264, −6.99845290625400236868305640594, −5.61468196509053665265226327032, −5.183871533977615501201876634647, −3.357788548270417245295888987380, −2.27154964558707562142099191894, −1.08751870479458982049399333428,
2.30900037012692884308597747827, 3.22981556356350304837525272311, 4.89022227371268990302701652710, 5.36687796034836479673832993984, 6.14164931409451993589058671731, 7.585990665074592069526994460, 9.004117348448925328944499535721, 9.93498630551973137113472432799, 10.97565825832079065722566323823, 12.30436590508331397651250259004, 12.75197365762527349423649361439, 14.20586649123004164450092487397, 14.90488192897150078684924740496, 15.79560292500051246322065177791, 16.63365207537198580241187292482, 17.469324540087230279412211413530, 18.44921366783219621813297302253, 20.227420820201651676652441500333, 20.97745570512639367673195987622, 21.69750735771223530253863985733, 22.27264302457114570253215120111, 23.127953993175141347392230460733, 24.30246400554054474496813452590, 25.09707290571398426167570232393, 25.935373901145167142416968539814