L(s) = 1 | + (0.0697 + 0.997i)2-s + (0.999 + 0.0348i)3-s + (−0.990 + 0.139i)4-s + (0.0348 + 0.999i)6-s + (−0.866 − 0.5i)7-s + (−0.207 − 0.978i)8-s + (0.997 + 0.0697i)9-s + (−0.104 + 0.994i)11-s + (−0.994 + 0.104i)12-s + (0.829 + 0.559i)13-s + (0.438 − 0.898i)14-s + (0.961 − 0.275i)16-s + (0.927 − 0.374i)17-s + i·18-s + (−0.848 − 0.529i)21-s + (−0.999 − 0.0348i)22-s + ⋯ |
L(s) = 1 | + (0.0697 + 0.997i)2-s + (0.999 + 0.0348i)3-s + (−0.990 + 0.139i)4-s + (0.0348 + 0.999i)6-s + (−0.866 − 0.5i)7-s + (−0.207 − 0.978i)8-s + (0.997 + 0.0697i)9-s + (−0.104 + 0.994i)11-s + (−0.994 + 0.104i)12-s + (0.829 + 0.559i)13-s + (0.438 − 0.898i)14-s + (0.961 − 0.275i)16-s + (0.927 − 0.374i)17-s + i·18-s + (−0.848 − 0.529i)21-s + (−0.999 − 0.0348i)22-s + ⋯ |
Λ(s)=(=(475s/2ΓR(s)L(s)(−0.161+0.986i)Λ(1−s)
Λ(s)=(=(475s/2ΓR(s)L(s)(−0.161+0.986i)Λ(1−s)
Degree: |
1 |
Conductor: |
475
= 52⋅19
|
Sign: |
−0.161+0.986i
|
Analytic conductor: |
2.20589 |
Root analytic conductor: |
2.20589 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ475(147,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 475, (0: ), −0.161+0.986i)
|
Particular Values
L(21) |
≈ |
1.096953537+1.291170169i |
L(21) |
≈ |
1.096953537+1.291170169i |
L(1) |
≈ |
1.130317191+0.7192136359i |
L(1) |
≈ |
1.130317191+0.7192136359i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1 |
| 19 | 1 |
good | 2 | 1+(0.0697+0.997i)T |
| 3 | 1+(0.999+0.0348i)T |
| 7 | 1+(−0.866−0.5i)T |
| 11 | 1+(−0.104+0.994i)T |
| 13 | 1+(0.829+0.559i)T |
| 17 | 1+(0.927−0.374i)T |
| 23 | 1+(0.694+0.719i)T |
| 29 | 1+(−0.374+0.927i)T |
| 31 | 1+(−0.669−0.743i)T |
| 37 | 1+(0.587+0.809i)T |
| 41 | 1+(−0.961+0.275i)T |
| 43 | 1+(0.984−0.173i)T |
| 47 | 1+(−0.927−0.374i)T |
| 53 | 1+(0.139+0.990i)T |
| 59 | 1+(−0.241+0.970i)T |
| 61 | 1+(−0.719+0.694i)T |
| 67 | 1+(−0.529−0.848i)T |
| 71 | 1+(0.882−0.469i)T |
| 73 | 1+(0.829−0.559i)T |
| 79 | 1+(0.0348−0.999i)T |
| 83 | 1+(0.743−0.669i)T |
| 89 | 1+(0.961+0.275i)T |
| 97 | 1+(0.529−0.848i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−23.40001463373897584597967949723, −22.574613899820771668450917099661, −21.57087208807053361484811171938, −21.05127084949523791794161363389, −20.18580196232181493326232159620, −19.256304485363139891073937464583, −18.85891652768281244257115281853, −18.100107022570621037372934146106, −16.65817776342510119965856803127, −15.70554646601963378740609899704, −14.69566270660942364608636914594, −13.865276286008835628825478202175, −12.976360893175867710332972347320, −12.54904305282141392604302983335, −11.20798752694527655672332586345, −10.31233425175295733005224765726, −9.42595499414099065933080295292, −8.65431693992369876096621867161, −7.93245480777772725001994268051, −6.326026657459784854476358553461, −5.29096920756857327482676839857, −3.73680758594325288929287733147, −3.27886765680325856833246338566, −2.32281981631292686683463607840, −0.95446375228598719252223085859,
1.38919290145519021995607035842, 3.130467921307834510357211504011, 3.88640322060920661628689968933, 4.913851995248517395825410515832, 6.26943778196027955790400142953, 7.214943915393773630002696293468, 7.741354501363980914538052830177, 9.05912491581363157442727011753, 9.50815616112200526313678932961, 10.46471872791159582392878620660, 12.22971643209601874165710036581, 13.19943931177412626907037106590, 13.65411197315391053622915050440, 14.69168983930552415982467740473, 15.33181693600992988643339358427, 16.26563741054596999794398254665, 16.87590226426984214543757991407, 18.21982706090848766180445048934, 18.7776418550411220437186184601, 19.75354889824837637728984255571, 20.63913382154884925252338655983, 21.54205758469719294503860058154, 22.608846497967784864451669999293, 23.357389689890666435548896284054, 24.04815831093765428997144377552