L(s) = 1 | + (0.990 + 0.139i)2-s + (−0.997 + 0.0697i)3-s + (0.961 + 0.275i)4-s + (−0.997 − 0.0697i)6-s + (−0.5 + 0.866i)7-s + (0.913 + 0.406i)8-s + (0.990 − 0.139i)9-s + (−0.978 + 0.207i)11-s + (−0.978 − 0.207i)12-s + (−0.374 + 0.927i)13-s + (−0.615 + 0.788i)14-s + (0.848 + 0.529i)16-s + (−0.719 − 0.694i)17-s + 18-s + (0.438 − 0.898i)21-s + (−0.997 + 0.0697i)22-s + ⋯ |
L(s) = 1 | + (0.990 + 0.139i)2-s + (−0.997 + 0.0697i)3-s + (0.961 + 0.275i)4-s + (−0.997 − 0.0697i)6-s + (−0.5 + 0.866i)7-s + (0.913 + 0.406i)8-s + (0.990 − 0.139i)9-s + (−0.978 + 0.207i)11-s + (−0.978 − 0.207i)12-s + (−0.374 + 0.927i)13-s + (−0.615 + 0.788i)14-s + (0.848 + 0.529i)16-s + (−0.719 − 0.694i)17-s + 18-s + (0.438 − 0.898i)21-s + (−0.997 + 0.0697i)22-s + ⋯ |
Λ(s)=(=(475s/2ΓR(s)L(s)(−0.560+0.828i)Λ(1−s)
Λ(s)=(=(475s/2ΓR(s)L(s)(−0.560+0.828i)Λ(1−s)
Degree: |
1 |
Conductor: |
475
= 52⋅19
|
Sign: |
−0.560+0.828i
|
Analytic conductor: |
2.20589 |
Root analytic conductor: |
2.20589 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ475(111,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 475, (0: ), −0.560+0.828i)
|
Particular Values
L(21) |
≈ |
0.5872289386+1.106004444i |
L(21) |
≈ |
0.5872289386+1.106004444i |
L(1) |
≈ |
1.083743782+0.4757277348i |
L(1) |
≈ |
1.083743782+0.4757277348i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1 |
| 19 | 1 |
good | 2 | 1+(0.990+0.139i)T |
| 3 | 1+(−0.997+0.0697i)T |
| 7 | 1+(−0.5+0.866i)T |
| 11 | 1+(−0.978+0.207i)T |
| 13 | 1+(−0.374+0.927i)T |
| 17 | 1+(−0.719−0.694i)T |
| 23 | 1+(0.0348+0.999i)T |
| 29 | 1+(−0.719+0.694i)T |
| 31 | 1+(−0.104−0.994i)T |
| 37 | 1+(0.309+0.951i)T |
| 41 | 1+(0.848+0.529i)T |
| 43 | 1+(−0.939−0.342i)T |
| 47 | 1+(−0.719+0.694i)T |
| 53 | 1+(0.961+0.275i)T |
| 59 | 1+(−0.882+0.469i)T |
| 61 | 1+(0.0348+0.999i)T |
| 67 | 1+(0.438+0.898i)T |
| 71 | 1+(0.559+0.829i)T |
| 73 | 1+(−0.374−0.927i)T |
| 79 | 1+(−0.997+0.0697i)T |
| 83 | 1+(−0.104−0.994i)T |
| 89 | 1+(0.848−0.529i)T |
| 97 | 1+(0.438−0.898i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−23.10252430632814686198872780277, −22.99829128953046432464352297037, −21.9694840483993133480700097751, −21.23575873340530894121060442529, −20.25144359050944439519949825071, −19.488906880824238819379515880213, −18.36952328519543144908236253725, −17.34257889282611868796227028316, −16.50714887366362814906307078334, −15.78986277867430611856688545809, −14.955265666861824818276546479723, −13.694056110695864399377790173008, −12.86895283537075255129285878287, −12.52559222897192085388379557928, −11.1301165414991474526505585053, −10.63043791191978410815974322068, −9.92275533042105253557260456419, −7.96821384022877678472524876942, −7.05340412693688710928545430616, −6.2170292651751358038797026270, −5.31968445658655096539427994036, −4.44760918316308838624094448879, −3.41980682152757144201135261609, −2.11203758073882374793944633501, −0.52473755973612859345777161286,
1.852103563488171271116064563299, 2.91580596772945985757688816701, 4.28987115396611531639884032313, 5.13128065864817964629668075474, 5.87502311870162551171715748382, 6.79654862305847374044937748582, 7.616951489066477467969719993177, 9.22522820818433543672119852767, 10.22366795916987360466953074922, 11.4395466393824993808911812118, 11.77211057084563925545422215588, 12.91133194667832780109507909259, 13.35000749754730115969542584667, 14.78471908118875242699740152043, 15.57883851668829969596124470804, 16.17015628788471303242516912041, 16.99009920407620172506931315236, 18.12730322259652654682276054963, 18.89380420595908462700577519663, 20.065428197156791628247660023057, 21.16614323320456450810528781458, 21.742839789115570586120676797331, 22.42850588910634229657160324851, 23.173555755852406176952369427821, 24.00557708612125526483090085923