L(s) = 1 | + (−1.28 − 2.21i)2-s + (1.84 + 3.19i)3-s + (0.719 − 1.24i)4-s + 0.561·5-s + (4.71 − 8.17i)6-s + (−9.08 + 15.7i)7-s − 24.1·8-s + (6.71 − 11.6i)9-s + (−0.719 − 1.24i)10-s + (−32.3 − 56.0i)11-s + 5.30·12-s + 46.5·14-s + (1.03 + 1.79i)15-s + (25.2 + 43.6i)16-s + (12.7 − 22.1i)17-s − 34.3·18-s + ⋯ |
L(s) = 1 | + (−0.452 − 0.784i)2-s + (0.354 + 0.614i)3-s + (0.0899 − 0.155i)4-s + 0.0502·5-s + (0.321 − 0.556i)6-s + (−0.490 + 0.849i)7-s − 1.06·8-s + (0.248 − 0.430i)9-s + (−0.0227 − 0.0393i)10-s + (−0.887 − 1.53i)11-s + 0.127·12-s + 0.888·14-s + (0.0178 + 0.0308i)15-s + (0.393 + 0.682i)16-s + (0.182 − 0.315i)17-s − 0.450·18-s + ⋯ |
Λ(s)=(=(169s/2ΓC(s)L(s)(−0.872+0.488i)Λ(4−s)
Λ(s)=(=(169s/2ΓC(s+3/2)L(s)(−0.872+0.488i)Λ(1−s)
Degree: |
2 |
Conductor: |
169
= 132
|
Sign: |
−0.872+0.488i
|
Analytic conductor: |
9.97132 |
Root analytic conductor: |
3.15774 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ169(146,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 169, ( :3/2), −0.872+0.488i)
|
Particular Values
L(2) |
≈ |
0.231529−0.886783i |
L(21) |
≈ |
0.231529−0.886783i |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 13 | 1 |
good | 2 | 1+(1.28+2.21i)T+(−4+6.92i)T2 |
| 3 | 1+(−1.84−3.19i)T+(−13.5+23.3i)T2 |
| 5 | 1−0.561T+125T2 |
| 7 | 1+(9.08−15.7i)T+(−171.5−297.i)T2 |
| 11 | 1+(32.3+56.0i)T+(−665.5+1.15e3i)T2 |
| 17 | 1+(−12.7+22.1i)T+(−2.45e3−4.25e3i)T2 |
| 19 | 1+(−53.9+93.5i)T+(−3.42e3−5.94e3i)T2 |
| 23 | 1+(36.6+63.4i)T+(−6.08e3+1.05e4i)T2 |
| 29 | 1+(87.9+152.i)T+(−1.21e4+2.11e4i)T2 |
| 31 | 1+113.T+2.97e4T2 |
| 37 | 1+(57.4+99.4i)T+(−2.53e4+4.38e4i)T2 |
| 41 | 1+(−34.8−60.3i)T+(−3.44e4+5.96e4i)T2 |
| 43 | 1+(219.−379.i)T+(−3.97e4−6.88e4i)T2 |
| 47 | 1+31.9T+1.03e5T2 |
| 53 | 1−2.84T+1.48e5T2 |
| 59 | 1+(35.8−62.0i)T+(−1.02e5−1.77e5i)T2 |
| 61 | 1+(−460.+797.i)T+(−1.13e5−1.96e5i)T2 |
| 67 | 1+(−222.−384.i)T+(−1.50e5+2.60e5i)T2 |
| 71 | 1+(−270.+469.i)T+(−1.78e5−3.09e5i)T2 |
| 73 | 1−764.T+3.89e5T2 |
| 79 | 1+421.T+4.93e5T2 |
| 83 | 1−603.T+5.71e5T2 |
| 89 | 1+(−579.−1.00e3i)T+(−3.52e5+6.10e5i)T2 |
| 97 | 1+(291.−505.i)T+(−4.56e5−7.90e5i)T2 |
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L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−11.61497385325112472318886278447, −10.88584368082938731139863261068, −9.724504376464827791275988995464, −9.270299142840876560129056485539, −8.208238680157439429258784355755, −6.37051195911218099393106461508, −5.39699266518487194503548511801, −3.44947885884092282924320216215, −2.54321628636650256424530971474, −0.44194807365249133003897886939,
1.95652601494146955328175586963, 3.65808196524719961611417011260, 5.44835332201661153865513189541, 6.99527308058620390429088172851, 7.42982581603841182947961645868, 8.195272314577383850875855156911, 9.671714918396222049874846998149, 10.40122851724185191646898306236, 12.05048192882306585109859818240, 12.80610647486775650638328882253