L(s) = 1 | + (−0.831 − 0.555i)3-s + (−0.980 − 0.195i)5-s + (−0.382 + 0.923i)7-s + (0.382 + 0.923i)9-s + (0.555 + 0.831i)11-s + (0.980 − 0.195i)13-s + (0.707 + 0.707i)15-s + (0.707 − 0.707i)17-s + (0.195 + 0.980i)19-s + (0.831 − 0.555i)21-s + (−0.923 + 0.382i)23-s + (0.923 + 0.382i)25-s + (0.195 − 0.980i)27-s + (−0.555 + 0.831i)29-s + i·31-s + ⋯ |
L(s) = 1 | + (−0.831 − 0.555i)3-s + (−0.980 − 0.195i)5-s + (−0.382 + 0.923i)7-s + (0.382 + 0.923i)9-s + (0.555 + 0.831i)11-s + (0.980 − 0.195i)13-s + (0.707 + 0.707i)15-s + (0.707 − 0.707i)17-s + (0.195 + 0.980i)19-s + (0.831 − 0.555i)21-s + (−0.923 + 0.382i)23-s + (0.923 + 0.382i)25-s + (0.195 − 0.980i)27-s + (−0.555 + 0.831i)29-s + i·31-s + ⋯ |
Λ(s)=(=(128s/2ΓR(s)L(s)(0.671+0.740i)Λ(1−s)
Λ(s)=(=(128s/2ΓR(s)L(s)(0.671+0.740i)Λ(1−s)
Degree: |
1 |
Conductor: |
128
= 27
|
Sign: |
0.671+0.740i
|
Analytic conductor: |
0.594429 |
Root analytic conductor: |
0.594429 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ128(69,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 128, (0: ), 0.671+0.740i)
|
Particular Values
L(21) |
≈ |
0.5705729636+0.2529176002i |
L(21) |
≈ |
0.5705729636+0.2529176002i |
L(1) |
≈ |
0.6976426151+0.06235495933i |
L(1) |
≈ |
0.6976426151+0.06235495933i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
good | 3 | 1+(−0.831−0.555i)T |
| 5 | 1+(−0.980−0.195i)T |
| 7 | 1+(−0.382+0.923i)T |
| 11 | 1+(0.555+0.831i)T |
| 13 | 1+(0.980−0.195i)T |
| 17 | 1+(0.707−0.707i)T |
| 19 | 1+(0.195+0.980i)T |
| 23 | 1+(−0.923+0.382i)T |
| 29 | 1+(−0.555+0.831i)T |
| 31 | 1+iT |
| 37 | 1+(−0.195+0.980i)T |
| 41 | 1+(0.923−0.382i)T |
| 43 | 1+(−0.831+0.555i)T |
| 47 | 1+(−0.707+0.707i)T |
| 53 | 1+(−0.555−0.831i)T |
| 59 | 1+(0.980+0.195i)T |
| 61 | 1+(0.831+0.555i)T |
| 67 | 1+(0.831+0.555i)T |
| 71 | 1+(0.382−0.923i)T |
| 73 | 1+(−0.382−0.923i)T |
| 79 | 1+(−0.707−0.707i)T |
| 83 | 1+(−0.195−0.980i)T |
| 89 | 1+(−0.923−0.382i)T |
| 97 | 1+iT |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−28.336730512250287394627405071465, −27.81664565432787162302838948791, −26.587743038612730323918972412823, −26.207074472579336238363039873757, −24.24558879705583406424564728913, −23.49496516899416746357633174615, −22.750285639855725544860603990184, −21.77611450575634261921903980608, −20.57778747199566542238476429258, −19.52722707029615052623007869362, −18.49079319859548850314650535305, −17.10050100976301413451689338492, −16.33779123211096050865825424380, −15.53801968850239494225413010737, −14.231009231329941372953344954254, −12.87011964663447448591377372023, −11.510672661819755696445893664281, −10.97937961874155583897847727926, −9.77168956325543050707345743357, −8.30608680528892295148457695616, −6.91233050935855536464811722987, −5.88911634285213075419714890494, −4.143394214901827843255250879826, −3.605266093801941166130619865077, −0.7140732234628200394748478401,
1.50042838544100525252861036693, 3.47111303211322338592640657412, 4.99887080329018367959001748410, 6.16050110473938915286046851287, 7.35294959093049726900898431260, 8.46660191792783481324382963447, 9.95020125366921859980407641928, 11.44367352619588057908616333286, 12.102710194443507951678321447610, 12.88872110936633783841813610803, 14.49685995585264634660855877399, 15.86623868088130510472098215563, 16.3863111599850482719156609161, 17.89553589435387612816161335906, 18.63991165511321091550264844542, 19.587680615445968039975477585060, 20.78952001301146099524759506462, 22.28854046603295475308686383406, 22.89779144190730880975773594280, 23.75916801357984521803666238116, 24.893470225628608174103301998674, 25.65205142360129177192487311609, 27.4588938764957538084999238810, 27.87242623369062871836775329852, 28.76826371817605490473444314993