L(s) = 1 | + (−0.961 − 0.275i)2-s + (0.848 + 0.529i)4-s + (−0.766 − 0.642i)7-s + (−0.669 − 0.743i)8-s + (−0.719 + 0.694i)11-s + (−0.961 + 0.275i)13-s + (0.559 + 0.829i)14-s + (0.438 + 0.898i)16-s + (0.978 − 0.207i)17-s + (0.669 + 0.743i)19-s + (0.882 − 0.469i)22-s + (0.997 − 0.0697i)23-s + 26-s + (−0.309 − 0.951i)28-s + (−0.615 − 0.788i)29-s + ⋯ |
L(s) = 1 | + (−0.961 − 0.275i)2-s + (0.848 + 0.529i)4-s + (−0.766 − 0.642i)7-s + (−0.669 − 0.743i)8-s + (−0.719 + 0.694i)11-s + (−0.961 + 0.275i)13-s + (0.559 + 0.829i)14-s + (0.438 + 0.898i)16-s + (0.978 − 0.207i)17-s + (0.669 + 0.743i)19-s + (0.882 − 0.469i)22-s + (0.997 − 0.0697i)23-s + 26-s + (−0.309 − 0.951i)28-s + (−0.615 − 0.788i)29-s + ⋯ |
Λ(s)=(=(675s/2ΓR(s)L(s)(−0.280−0.959i)Λ(1−s)
Λ(s)=(=(675s/2ΓR(s)L(s)(−0.280−0.959i)Λ(1−s)
Degree: |
1 |
Conductor: |
675
= 33⋅52
|
Sign: |
−0.280−0.959i
|
Analytic conductor: |
3.13468 |
Root analytic conductor: |
3.13468 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ675(229,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 675, (0: ), −0.280−0.959i)
|
Particular Values
L(21) |
≈ |
0.3047804080−0.4064216225i |
L(21) |
≈ |
0.3047804080−0.4064216225i |
L(1) |
≈ |
0.5589514212−0.1348842514i |
L(1) |
≈ |
0.5589514212−0.1348842514i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 5 | 1 |
good | 2 | 1+(−0.961−0.275i)T |
| 7 | 1+(−0.766−0.642i)T |
| 11 | 1+(−0.719+0.694i)T |
| 13 | 1+(−0.961+0.275i)T |
| 17 | 1+(0.978−0.207i)T |
| 19 | 1+(0.669+0.743i)T |
| 23 | 1+(0.997−0.0697i)T |
| 29 | 1+(−0.615−0.788i)T |
| 31 | 1+(−0.374−0.927i)T |
| 37 | 1+(−0.913−0.406i)T |
| 41 | 1+(0.961−0.275i)T |
| 43 | 1+(−0.173+0.984i)T |
| 47 | 1+(0.374−0.927i)T |
| 53 | 1+(−0.309−0.951i)T |
| 59 | 1+(−0.719−0.694i)T |
| 61 | 1+(−0.241−0.970i)T |
| 67 | 1+(0.615−0.788i)T |
| 71 | 1+(0.669−0.743i)T |
| 73 | 1+(−0.913+0.406i)T |
| 79 | 1+(−0.615−0.788i)T |
| 83 | 1+(−0.0348−0.999i)T |
| 89 | 1+(0.913−0.406i)T |
| 97 | 1+(−0.990+0.139i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−23.1395870539330754114481352162, −22.07026915687694178914448654555, −21.31288115392233238973903254369, −20.34816623416795591615839958195, −19.43895404487620687396145308103, −18.910912179173659885126240484449, −18.181221193363929936119071088681, −17.222505526790065663065672645803, −16.44190192716360621308962819806, −15.73003617774106458824709245850, −15.02514419239059381281851886405, −14.04414593658951414106777071216, −12.79980052238835490932683077211, −12.082761819112459741497781979, −10.99440134605465433111908594643, −10.22854967362588214029906961032, −9.334965031081699701160444228864, −8.69591409541768543901153704287, −7.58116724482886400647827798805, −6.93399110069232732572461187513, −5.69137952999830684916177718700, −5.214127468059619194711124421302, −3.17340412484624597051045537621, −2.64367618583787976284536430954, −1.11403418747516941067059202354,
0.377765650094207629114563477398, 1.80743901853855722796635856581, 2.88128394076324538468108571744, 3.79405089784161041426609912171, 5.175383158858211394475977644810, 6.39163067715567231882631618682, 7.52560119295411235321144534284, 7.65647856150428081841012197659, 9.24825771835173357524196274075, 9.80848723385930426491938752241, 10.431426054372766469051299303442, 11.49992254257665131922500468741, 12.43840095022062947887226618512, 13.0312641507818199608823495044, 14.30703253612525164193552043745, 15.283891652996867699434716964078, 16.19243233696779705404743020328, 16.86289209379940141540796372809, 17.509528873996813202787077648007, 18.646133984001188970813807894423, 19.07682356758589390389011404892, 20.05372817050168367116282095457, 20.663036769442183379710241884963, 21.413490743187805864294697657, 22.59929177832188146247307867878