L(s) = 1 | + (−0.754 − 0.656i)2-s + (−0.592 + 0.805i)3-s + (0.137 + 0.990i)4-s + (−0.962 + 0.272i)5-s + (0.975 − 0.218i)6-s + (0.789 + 0.614i)7-s + (0.546 − 0.837i)8-s + (−0.298 − 0.954i)9-s + (0.904 + 0.426i)10-s + (−0.677 − 0.735i)11-s + (−0.879 − 0.475i)12-s + (0.993 + 0.110i)13-s + (−0.191 − 0.981i)14-s + (0.350 − 0.936i)15-s + (−0.962 + 0.272i)16-s + (0.137 − 0.990i)17-s + ⋯ |
L(s) = 1 | + (−0.754 − 0.656i)2-s + (−0.592 + 0.805i)3-s + (0.137 + 0.990i)4-s + (−0.962 + 0.272i)5-s + (0.975 − 0.218i)6-s + (0.789 + 0.614i)7-s + (0.546 − 0.837i)8-s + (−0.298 − 0.954i)9-s + (0.904 + 0.426i)10-s + (−0.677 − 0.735i)11-s + (−0.879 − 0.475i)12-s + (0.993 + 0.110i)13-s + (−0.191 − 0.981i)14-s + (0.350 − 0.936i)15-s + (−0.962 + 0.272i)16-s + (0.137 − 0.990i)17-s + ⋯ |
Λ(s)=(=(361s/2ΓR(s)L(s)(0.993−0.115i)Λ(1−s)
Λ(s)=(=(361s/2ΓR(s)L(s)(0.993−0.115i)Λ(1−s)
Degree: |
1 |
Conductor: |
361
= 192
|
Sign: |
0.993−0.115i
|
Analytic conductor: |
1.67647 |
Root analytic conductor: |
1.67647 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ361(163,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 361, (0: ), 0.993−0.115i)
|
Particular Values
L(21) |
≈ |
0.5952426370−0.03457268367i |
L(21) |
≈ |
0.5952426370−0.03457268367i |
L(1) |
≈ |
0.5821154419+0.005771500786i |
L(1) |
≈ |
0.5821154419+0.005771500786i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 19 | 1 |
good | 2 | 1+(−0.754−0.656i)T |
| 3 | 1+(−0.592+0.805i)T |
| 5 | 1+(−0.962+0.272i)T |
| 7 | 1+(0.789+0.614i)T |
| 11 | 1+(−0.677−0.735i)T |
| 13 | 1+(0.993+0.110i)T |
| 17 | 1+(0.137−0.990i)T |
| 23 | 1+(−0.592−0.805i)T |
| 29 | 1+(−0.926+0.376i)T |
| 31 | 1+(0.945+0.324i)T |
| 37 | 1+(−0.677−0.735i)T |
| 41 | 1+(0.635+0.771i)T |
| 43 | 1+(0.904−0.426i)T |
| 47 | 1+(0.975−0.218i)T |
| 53 | 1+(0.975−0.218i)T |
| 59 | 1+(0.635+0.771i)T |
| 61 | 1+(−0.998+0.0550i)T |
| 67 | 1+(0.451+0.892i)T |
| 71 | 1+(−0.998−0.0550i)T |
| 73 | 1+(0.137−0.990i)T |
| 79 | 1+(0.904−0.426i)T |
| 83 | 1+(0.245+0.969i)T |
| 89 | 1+(0.137+0.990i)T |
| 97 | 1+(0.451−0.892i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−24.44546057302590508593416830256, −23.95302721572895943268006625420, −23.321325416182266738816251394131, −22.72818143073819881828340996520, −20.89840192415856070993292060900, −20.09802678804730267305636100036, −19.18989352096198963378966993576, −18.45027063876942369184291021363, −17.543641962650369314403211679985, −17.00350173899211150104002403707, −15.89619604552095410398511452541, −15.223143250185637634654857158655, −14.00789167938213134699254057366, −13.04744822708421001292797153237, −11.85324178682116248900009731201, −11.0150913665133324408439663351, −10.31370702175249278949633625466, −8.67059468487354983263019106174, −7.7755222465014613169108111221, −7.48632470242345272380823379259, −6.20635633157006160620453486035, −5.19903917320616084473239177705, −4.10997246193790297618540567562, −1.93094563417554865132639948716, −0.89977560610716094463116762153,
0.74536107080570370010964670024, 2.61679448025103715500210651932, 3.64790791201515348670026287376, 4.61899337368629243199289280750, 5.88669278923133044499630807369, 7.32366727818867097223688262318, 8.4109818540013375086860702749, 8.99564367529108027176451164706, 10.4099435589260880310336661529, 11.03792655442407397477479199729, 11.65766255960438311112125701844, 12.42508145385626617621146081663, 13.95383056594333872780305636854, 15.2598087174479589024238906100, 16.019504641314860154528525591442, 16.54482594461676919751316121579, 18.00139493830572139902363663363, 18.32897289951662458718145352558, 19.30763720625174449666393972273, 20.68528689569828247314352701248, 20.865180072440351255889746859, 21.96429235842975024725072896800, 22.724444607415057058122799421590, 23.69774699778005810786083323407, 24.7332522657898231280100228190