L(s) = 1 | + (0.766 − 0.642i)2-s + (0.173 − 0.984i)4-s + (0.173 − 0.984i)5-s + (−0.5 − 0.866i)8-s + (−0.5 − 0.866i)10-s + (0.173 + 0.984i)11-s + (0.173 − 0.984i)13-s + (−0.939 − 0.342i)16-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.939 − 0.342i)20-s + (0.766 + 0.642i)22-s + (0.766 + 0.642i)23-s + (−0.939 − 0.342i)25-s + (−0.5 − 0.866i)26-s + ⋯ |
L(s) = 1 | + (0.766 − 0.642i)2-s + (0.173 − 0.984i)4-s + (0.173 − 0.984i)5-s + (−0.5 − 0.866i)8-s + (−0.5 − 0.866i)10-s + (0.173 + 0.984i)11-s + (0.173 − 0.984i)13-s + (−0.939 − 0.342i)16-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.939 − 0.342i)20-s + (0.766 + 0.642i)22-s + (0.766 + 0.642i)23-s + (−0.939 − 0.342i)25-s + (−0.5 − 0.866i)26-s + ⋯ |
Λ(s)=(=(189s/2ΓR(s)L(s)(−0.337−0.941i)Λ(1−s)
Λ(s)=(=(189s/2ΓR(s)L(s)(−0.337−0.941i)Λ(1−s)
Degree: |
1 |
Conductor: |
189
= 33⋅7
|
Sign: |
−0.337−0.941i
|
Analytic conductor: |
0.877712 |
Root analytic conductor: |
0.877712 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ189(16,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 189, (0: ), −0.337−0.941i)
|
Particular Values
L(21) |
≈ |
0.9778795499−1.388860743i |
L(21) |
≈ |
0.9778795499−1.388860743i |
L(1) |
≈ |
1.242404790−0.8836570766i |
L(1) |
≈ |
1.242404790−0.8836570766i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 7 | 1 |
good | 2 | 1+(0.766−0.642i)T |
| 5 | 1+(0.173−0.984i)T |
| 11 | 1+(0.173+0.984i)T |
| 13 | 1+(0.173−0.984i)T |
| 17 | 1+(−0.5−0.866i)T |
| 19 | 1+(−0.5+0.866i)T |
| 23 | 1+(0.766+0.642i)T |
| 29 | 1+(0.173+0.984i)T |
| 31 | 1+(0.173−0.984i)T |
| 37 | 1+T |
| 41 | 1+(0.173−0.984i)T |
| 43 | 1+(0.766−0.642i)T |
| 47 | 1+(0.173+0.984i)T |
| 53 | 1+(−0.5+0.866i)T |
| 59 | 1+(−0.939+0.342i)T |
| 61 | 1+(0.173+0.984i)T |
| 67 | 1+(0.766+0.642i)T |
| 71 | 1+(−0.5+0.866i)T |
| 73 | 1+T |
| 79 | 1+(0.766−0.642i)T |
| 83 | 1+(0.173+0.984i)T |
| 89 | 1+(−0.5+0.866i)T |
| 97 | 1+(0.766−0.642i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−26.7938768215222076516057480874, −26.49773515401945196264003469719, −25.48124823297552136804491708973, −24.46466954111224080550503915018, −23.59601763050005382203063494230, −22.737340592196962667866447126357, −21.54103143127584821544247659147, −21.46207782201514571047144935818, −19.6770555099990712526404770641, −18.69508566347167923502293090426, −17.54108410168715869166235557278, −16.64256554913019053191740065042, −15.54319874525861882336126837378, −14.643915122917117155987058105011, −13.84654446309521755137795039188, −12.94723995551032277599800187809, −11.503355185758465470303982208700, −10.8527991156861656526585011117, −9.12590508673348549177453160930, −8.03182884311637492670623517077, −6.615672816980227364739536795408, −6.26321898595358571269328662395, −4.65814181234681626274527062424, −3.49529880201091441396534120586, −2.34936775154321311659227649360,
1.15562889878905185003921341735, 2.48980978877575016307727382733, 4.01654045626068712265621739933, 4.9830535540454793093875598592, 5.94133507371473984620169951042, 7.44779189496834642022348592182, 9.00863359837715428281414124288, 9.89350341483754042663657522359, 11.05291485258063268501566498515, 12.29216510428448477442567069863, 12.84052577606663470276065840217, 13.83777365049080520437500553475, 15.03783196964532073132726525973, 15.876959569995209828110065705680, 17.181834630918201682223425545625, 18.235963376916538413160530845763, 19.51417799341831085083183355205, 20.45737718205507224978427742371, 20.81383819265117754675130690455, 22.11451402249225731711651583308, 22.95537740470565000593512575251, 23.808053843198760126762010145871, 24.92014435527692078423951736728, 25.424920567567810434100673514365, 27.40406018832863766940898322390