L(s) = 1 | + (−1.67 − 2.90i)2-s + (−1.29 − 2.70i)3-s + (−3.62 + 6.28i)4-s + (0.769 + 0.444i)5-s + (−5.68 + 8.30i)6-s + 10.9·8-s + (−5.64 + 7.01i)9-s − 2.98i·10-s + (2.75 + 4.76i)11-s + (21.7 + 1.67i)12-s + (12.3 + 7.11i)13-s + (0.205 − 2.65i)15-s + (−3.81 − 6.61i)16-s − 13.0i·17-s + (29.8 + 4.63i)18-s + 26.2i·19-s + ⋯ |
L(s) = 1 | + (−0.838 − 1.45i)2-s + (−0.431 − 0.901i)3-s + (−0.907 + 1.57i)4-s + (0.153 + 0.0888i)5-s + (−0.948 + 1.38i)6-s + 1.36·8-s + (−0.627 + 0.778i)9-s − 0.298i·10-s + (0.250 + 0.433i)11-s + (1.80 + 0.139i)12-s + (0.948 + 0.547i)13-s + (0.0136 − 0.177i)15-s + (−0.238 − 0.413i)16-s − 0.766i·17-s + (1.65 + 0.257i)18-s + 1.38i·19-s + ⋯ |
Λ(s)=(=(441s/2ΓC(s)L(s)(−0.307+0.951i)Λ(3−s)
Λ(s)=(=(441s/2ΓC(s+1)L(s)(−0.307+0.951i)Λ(1−s)
Degree: |
2 |
Conductor: |
441
= 32⋅72
|
Sign: |
−0.307+0.951i
|
Analytic conductor: |
12.0163 |
Root analytic conductor: |
3.46646 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ441(391,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 441, ( :1), −0.307+0.951i)
|
Particular Values
L(23) |
≈ |
0.517605−0.711620i |
L(21) |
≈ |
0.517605−0.711620i |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(1.29+2.70i)T |
| 7 | 1 |
good | 2 | 1+(1.67+2.90i)T+(−2+3.46i)T2 |
| 5 | 1+(−0.769−0.444i)T+(12.5+21.6i)T2 |
| 11 | 1+(−2.75−4.76i)T+(−60.5+104.i)T2 |
| 13 | 1+(−12.3−7.11i)T+(84.5+146.i)T2 |
| 17 | 1+13.0iT−289T2 |
| 19 | 1−26.2iT−361T2 |
| 23 | 1+(−18.0+31.2i)T+(−264.5−458.i)T2 |
| 29 | 1+(−16.4−28.5i)T+(−420.5+728.i)T2 |
| 31 | 1+(−32.8−18.9i)T+(480.5+832.i)T2 |
| 37 | 1+42.9T+1.36e3T2 |
| 41 | 1+(−51.1−29.5i)T+(840.5+1.45e3i)T2 |
| 43 | 1+(14.4+25.0i)T+(−924.5+1.60e3i)T2 |
| 47 | 1+(10.8−6.27i)T+(1.10e3−1.91e3i)T2 |
| 53 | 1−9.76T+2.80e3T2 |
| 59 | 1+(−21.3−12.3i)T+(1.74e3+3.01e3i)T2 |
| 61 | 1+(15.7−9.08i)T+(1.86e3−3.22e3i)T2 |
| 67 | 1+(15.1−26.2i)T+(−2.24e3−3.88e3i)T2 |
| 71 | 1+43.4T+5.04e3T2 |
| 73 | 1−27.2iT−5.32e3T2 |
| 79 | 1+(−33.7−58.3i)T+(−3.12e3+5.40e3i)T2 |
| 83 | 1+(−8.30+4.79i)T+(3.44e3−5.96e3i)T2 |
| 89 | 1+63.3iT−7.92e3T2 |
| 97 | 1+(−76.9+44.4i)T+(4.70e3−8.14e3i)T2 |
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
−10.65668463896048501722791868425, −10.05858397500038476488942048494, −8.822243557182461173378754993429, −8.292155925734176719826273689342, −7.03979136139126957451473788507, −6.09370202275472909873008212802, −4.52593237766330329997523667541, −3.08015848259058070467435322153, −1.93065217344468899028971014495, −0.915266079448992479371087030423,
0.76318078386830230956626131047, 3.43983448861708490691627397744, 4.83882997408878378546122931973, 5.79335704621493938279388935816, 6.35566205503225673622992074126, 7.54849025996628202969313696442, 8.592666438476820622893886533344, 9.153403505922036773634083536078, 9.975715751974002676439307581980, 10.90723929498857349660234184131