L(s) = 1 | + (0.884 + 1.53i)2-s + (2.43 − 4.21i)4-s + (−6.52 − 11.3i)5-s + (−18.4 − 0.966i)7-s + 22.7·8-s + (11.5 − 19.9i)10-s + (−26.4 + 45.8i)11-s − 71.7·13-s + (−14.8 − 29.1i)14-s + (0.663 + 1.14i)16-s + (−53.2 + 92.2i)17-s + (−26.9 − 46.7i)19-s − 63.5·20-s − 93.6·22-s + (−9.32 − 16.1i)23-s + ⋯ |
L(s) = 1 | + (0.312 + 0.541i)2-s + (0.304 − 0.527i)4-s + (−0.583 − 1.01i)5-s + (−0.998 − 0.0522i)7-s + 1.00·8-s + (0.365 − 0.632i)10-s + (−0.725 + 1.25i)11-s − 1.52·13-s + (−0.284 − 0.557i)14-s + (0.0103 + 0.0179i)16-s + (−0.760 + 1.31i)17-s + (−0.325 − 0.563i)19-s − 0.710·20-s − 0.907·22-s + (−0.0845 − 0.146i)23-s + ⋯ |
Λ(s)=(=(189s/2ΓC(s)L(s)(−0.917+0.396i)Λ(4−s)
Λ(s)=(=(189s/2ΓC(s+3/2)L(s)(−0.917+0.396i)Λ(1−s)
Degree: |
2 |
Conductor: |
189
= 33⋅7
|
Sign: |
−0.917+0.396i
|
Analytic conductor: |
11.1513 |
Root analytic conductor: |
3.33936 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ189(109,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 189, ( :3/2), −0.917+0.396i)
|
Particular Values
L(2) |
≈ |
0.0766041−0.370287i |
L(21) |
≈ |
0.0766041−0.370287i |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 7 | 1+(18.4+0.966i)T |
good | 2 | 1+(−0.884−1.53i)T+(−4+6.92i)T2 |
| 5 | 1+(6.52+11.3i)T+(−62.5+108.i)T2 |
| 11 | 1+(26.4−45.8i)T+(−665.5−1.15e3i)T2 |
| 13 | 1+71.7T+2.19e3T2 |
| 17 | 1+(53.2−92.2i)T+(−2.45e3−4.25e3i)T2 |
| 19 | 1+(26.9+46.7i)T+(−3.42e3+5.94e3i)T2 |
| 23 | 1+(9.32+16.1i)T+(−6.08e3+1.05e4i)T2 |
| 29 | 1−261.T+2.43e4T2 |
| 31 | 1+(−61.1+105.i)T+(−1.48e4−2.57e4i)T2 |
| 37 | 1+(139.+240.i)T+(−2.53e4+4.38e4i)T2 |
| 41 | 1+31.3T+6.89e4T2 |
| 43 | 1+347.T+7.95e4T2 |
| 47 | 1+(271.+469.i)T+(−5.19e4+8.99e4i)T2 |
| 53 | 1+(−128.+222.i)T+(−7.44e4−1.28e5i)T2 |
| 59 | 1+(157.−273.i)T+(−1.02e5−1.77e5i)T2 |
| 61 | 1+(69.4+120.i)T+(−1.13e5+1.96e5i)T2 |
| 67 | 1+(198.−344.i)T+(−1.50e5−2.60e5i)T2 |
| 71 | 1+843.T+3.57e5T2 |
| 73 | 1+(−436.+755.i)T+(−1.94e5−3.36e5i)T2 |
| 79 | 1+(−277.−480.i)T+(−2.46e5+4.26e5i)T2 |
| 83 | 1+297.T+5.71e5T2 |
| 89 | 1+(51.2+88.7i)T+(−3.52e5+6.10e5i)T2 |
| 97 | 1+515.T+9.12e5T2 |
show more | |
show less | |
L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−12.03117790526865345270574285450, −10.43021214573361830256533125756, −9.855670987842234339027930140182, −8.512020916926113184814723248941, −7.33409423546335944296319309085, −6.50814090720511389805486470348, −5.06555736921552801193022990457, −4.38113608137037837158208576663, −2.21870132029935207926582761390, −0.13653547828827663157199457907,
2.81962157662477851083733532657, 3.11443697521656853295142199567, 4.74738847110645436140468926181, 6.53445561401830796139607866521, 7.28775059322326560208887257976, 8.358802285204833381904426191230, 9.928490817020079564447301034068, 10.72015124114656030898045870459, 11.65629299538665751281542355934, 12.33014701446284728240424902788