L(s) = 1 | + (−0.396 + 0.918i)2-s + (−0.686 − 0.727i)4-s + (−0.893 − 0.448i)5-s + (0.686 − 0.727i)7-s + (0.939 − 0.342i)8-s + (0.766 − 0.642i)10-s + (0.835 + 0.549i)11-s + (0.396 + 0.918i)14-s + (−0.0581 + 0.998i)16-s + (−0.939 − 0.342i)17-s + (−0.173 + 0.984i)19-s + (0.286 + 0.957i)20-s + (−0.835 + 0.549i)22-s + (−0.686 − 0.727i)23-s + (0.597 + 0.802i)25-s + ⋯ |
L(s) = 1 | + (−0.396 + 0.918i)2-s + (−0.686 − 0.727i)4-s + (−0.893 − 0.448i)5-s + (0.686 − 0.727i)7-s + (0.939 − 0.342i)8-s + (0.766 − 0.642i)10-s + (0.835 + 0.549i)11-s + (0.396 + 0.918i)14-s + (−0.0581 + 0.998i)16-s + (−0.939 − 0.342i)17-s + (−0.173 + 0.984i)19-s + (0.286 + 0.957i)20-s + (−0.835 + 0.549i)22-s + (−0.686 − 0.727i)23-s + (0.597 + 0.802i)25-s + ⋯ |
Λ(s)=(=(1053s/2ΓR(s)L(s)(0.761+0.647i)Λ(1−s)
Λ(s)=(=(1053s/2ΓR(s)L(s)(0.761+0.647i)Λ(1−s)
Degree: |
1 |
Conductor: |
1053
= 34⋅13
|
Sign: |
0.761+0.647i
|
Analytic conductor: |
4.89011 |
Root analytic conductor: |
4.89011 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1053(283,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 1053, (0: ), 0.761+0.647i)
|
Particular Values
L(21) |
≈ |
0.9378208814+0.3448313349i |
L(21) |
≈ |
0.9378208814+0.3448313349i |
L(1) |
≈ |
0.7709803377+0.2177577893i |
L(1) |
≈ |
0.7709803377+0.2177577893i |
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+(−0.893−0.448i)T |
| 7 | 1+(0.686−0.727i)T |
| 11 | 1+(0.835+0.549i)T |
| 17 | 1+(−0.939−0.342i)T |
| 19 | 1+(−0.173+0.984i)T |
| 23 | 1+(−0.686−0.727i)T |
| 29 | 1+(0.597+0.802i)T |
| 31 | 1+(0.686+0.727i)T |
| 37 | 1+(0.939+0.342i)T |
| 41 | 1+(−0.597+0.802i)T |
| 43 | 1+(−0.0581+0.998i)T |
| 47 | 1+(0.686−0.727i)T |
| 53 | 1+(−0.5−0.866i)T |
| 59 | 1+(0.835−0.549i)T |
| 61 | 1+(−0.286−0.957i)T |
| 67 | 1+(0.993+0.116i)T |
| 71 | 1+(−0.173−0.984i)T |
| 73 | 1+(0.939−0.342i)T |
| 79 | 1+(−0.993+0.116i)T |
| 83 | 1+(0.993−0.116i)T |
| 89 | 1+(−0.173+0.984i)T |
| 97 | 1+(0.835+0.549i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−21.60098309169300492471013157649, −20.49060091802362873063809881601, −19.760069553316819518652843668841, −19.183254137558936987190342011995, −18.53912545600016931117828850955, −17.63746763059249477600160950685, −17.11795847205257962021047986684, −15.83007316682794241170469393401, −15.26435273248443013365796272643, −14.22767682236872834715856317313, −13.49485896984101213702931250579, −12.38571450949886465459569669304, −11.58956950217180791097230434432, −11.35508935100698386912890923190, −10.452117635416768516078642249990, −9.29490027554709157694800762252, −8.61433284353751380994932687267, −7.97101155861263630895837305118, −6.98112877665043908962994518050, −5.831427521461385082499934447055, −4.473774951354341674999181228525, −3.998439386478035116614087241494, −2.83155360809384327192212260346, −2.086320737193342177784291117458, −0.750322970293837274114390164831,
0.809109942373181210409112261779, 1.77363933285754488029086910974, 3.67610864316940762822732684171, 4.50617345828863583049401666244, 4.93598906496871939863371660002, 6.42881877869931034240334022186, 6.965853203335228422087247682972, 8.039796081367258683993662125164, 8.35208595197889090268122094953, 9.40419252638127449467826157669, 10.29806233740512237727992301481, 11.190747270670060470930227675866, 12.07979184503164683831100686179, 13.01984469955509579051801241020, 14.07860820341258462913287147626, 14.61018683067898427498441687345, 15.392968928195271599371614895090, 16.31055360932203086806910526241, 16.75949959925907619735836276566, 17.65404889684180572000424984542, 18.26335828297948087182853677387, 19.29851861773542524615583574213, 20.05171978560654079291725728469, 20.423806668748795830880699455674, 21.78329363454484193550316498555