L(s) = 1 | + (0.396 − 0.918i)2-s + (−0.686 − 0.727i)4-s + (−0.835 + 0.549i)5-s + (−0.286 − 0.957i)7-s + (−0.939 + 0.342i)8-s + (0.173 + 0.984i)10-s + (−0.835 − 0.549i)11-s + (−0.993 − 0.116i)14-s + (−0.0581 + 0.998i)16-s + (0.173 + 0.984i)17-s + (−0.939 + 0.342i)19-s + (0.973 + 0.230i)20-s + (−0.835 + 0.549i)22-s + (−0.286 + 0.957i)23-s + (0.396 − 0.918i)25-s + ⋯ |
L(s) = 1 | + (0.396 − 0.918i)2-s + (−0.686 − 0.727i)4-s + (−0.835 + 0.549i)5-s + (−0.286 − 0.957i)7-s + (−0.939 + 0.342i)8-s + (0.173 + 0.984i)10-s + (−0.835 − 0.549i)11-s + (−0.993 − 0.116i)14-s + (−0.0581 + 0.998i)16-s + (0.173 + 0.984i)17-s + (−0.939 + 0.342i)19-s + (0.973 + 0.230i)20-s + (−0.835 + 0.549i)22-s + (−0.286 + 0.957i)23-s + (0.396 − 0.918i)25-s + ⋯ |
Λ(s)=(=(1053s/2ΓR(s)L(s)(0.997−0.0742i)Λ(1−s)
Λ(s)=(=(1053s/2ΓR(s)L(s)(0.997−0.0742i)Λ(1−s)
Degree: |
1 |
Conductor: |
1053
= 34⋅13
|
Sign: |
0.997−0.0742i
|
Analytic conductor: |
4.89011 |
Root analytic conductor: |
4.89011 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1053(256,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 1053, (0: ), 0.997−0.0742i)
|
Particular Values
L(21) |
≈ |
0.7927017229+0.02945676966i |
L(21) |
≈ |
0.7927017229+0.02945676966i |
L(1) |
≈ |
0.7639598494−0.3464779210i |
L(1) |
≈ |
0.7639598494−0.3464779210i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 13 | 1 |
good | 2 | 1+(0.396−0.918i)T |
| 5 | 1+(−0.835+0.549i)T |
| 7 | 1+(−0.286−0.957i)T |
| 11 | 1+(−0.835−0.549i)T |
| 17 | 1+(0.173+0.984i)T |
| 19 | 1+(−0.939+0.342i)T |
| 23 | 1+(−0.286+0.957i)T |
| 29 | 1+(0.597+0.802i)T |
| 31 | 1+(0.973−0.230i)T |
| 37 | 1+(0.766−0.642i)T |
| 41 | 1+(−0.993−0.116i)T |
| 43 | 1+(−0.835−0.549i)T |
| 47 | 1+(0.973+0.230i)T |
| 53 | 1+(−0.5−0.866i)T |
| 59 | 1+(−0.835+0.549i)T |
| 61 | 1+(0.973+0.230i)T |
| 67 | 1+(0.396+0.918i)T |
| 71 | 1+(0.766−0.642i)T |
| 73 | 1+(−0.939+0.342i)T |
| 79 | 1+(0.597+0.802i)T |
| 83 | 1+(0.597+0.802i)T |
| 89 | 1+(0.766+0.642i)T |
| 97 | 1+(0.893−0.448i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−21.659133315203937823749015401299, −20.86898153424795084141095775925, −20.11120269725708097059876340595, −18.87150126865869035885905043409, −18.5014931635935255884147443877, −17.46931553246843005768370783071, −16.63831925057763647080160321338, −15.81552476513245863011370398683, −15.44507332912996714466868868085, −14.75185170633614387403639864220, −13.61159933592044282703588797842, −12.838929736740380020011009499776, −12.20887051616303239773500454794, −11.58177441448315576075595315684, −10.1427055863978154892039374300, −9.19387256477221023739823400777, −8.3768858978927149698108449854, −7.873287489640447910380392128235, −6.81198930305845398644133293113, −6.030025458527297423007349102945, −4.783775960788805420489156949527, −4.67523604422878174711904071908, −3.25082099412952029722663558214, −2.41559501316370408772088727448, −0.37428225177118964299251300998,
0.908865328895285965968698871903, 2.240004215347327176168935281606, 3.33862936977010739072566946589, 3.815258293976315380357833502304, 4.7205914406972374571514065423, 5.88132332654578494701483280531, 6.76732096890399446919692102016, 7.90567139543797564957669850821, 8.54511547375030214063348231485, 9.93453711890508124059191628598, 10.52220215533283597896590096458, 11.03257816627349767623693475102, 11.96026530760224191163514641369, 12.798043667764141853577465005068, 13.51246291523297173655362638474, 14.289385533669502160159578735941, 15.108941042283771784037970993203, 15.85765677129070581684113584357, 16.89273232106539486706039693100, 17.82505605816865837057377117845, 18.75649994284517161771080737560, 19.31332271833814812854892801419, 19.87130245444905826936636789546, 20.71455605650712114512434037057, 21.5310527123335681309390278354