L(s) = 1 | + (−0.982 − 1.70i)3-s + (0.349 + 0.605i)5-s + 3.80i·7-s + (−0.430 + 0.744i)9-s + 2.16i·11-s + (−1.16 − 0.672i)13-s + (0.686 − 1.18i)15-s + (−1.89 − 3.28i)17-s + (−1.62 + 4.04i)19-s + (6.46 − 3.73i)21-s + (−4.89 − 2.82i)23-s + (2.25 − 3.90i)25-s − 4.20·27-s + (−8.65 − 4.99i)29-s − 7.76·31-s + ⋯ |
L(s) = 1 | + (−0.567 − 0.982i)3-s + (0.156 + 0.270i)5-s + 1.43i·7-s + (−0.143 + 0.248i)9-s + 0.653i·11-s + (−0.323 − 0.186i)13-s + (0.177 − 0.307i)15-s + (−0.459 − 0.796i)17-s + (−0.372 + 0.928i)19-s + (1.41 − 0.814i)21-s + (−1.01 − 0.588i)23-s + (0.451 − 0.781i)25-s − 0.809·27-s + (−1.60 − 0.927i)29-s − 1.39·31-s + ⋯ |
Λ(s)=(=(1216s/2ΓC(s)L(s)(−0.887−0.461i)Λ(2−s)
Λ(s)=(=(1216s/2ΓC(s+1/2)L(s)(−0.887−0.461i)Λ(1−s)
Degree: |
2 |
Conductor: |
1216
= 26⋅19
|
Sign: |
−0.887−0.461i
|
Analytic conductor: |
9.70980 |
Root analytic conductor: |
3.11605 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1216(639,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1216, ( :1/2), −0.887−0.461i)
|
Particular Values
L(1) |
≈ |
0.1052258449 |
L(21) |
≈ |
0.1052258449 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 19 | 1+(1.62−4.04i)T |
good | 3 | 1+(0.982+1.70i)T+(−1.5+2.59i)T2 |
| 5 | 1+(−0.349−0.605i)T+(−2.5+4.33i)T2 |
| 7 | 1−3.80iT−7T2 |
| 11 | 1−2.16iT−11T2 |
| 13 | 1+(1.16+0.672i)T+(6.5+11.2i)T2 |
| 17 | 1+(1.89+3.28i)T+(−8.5+14.7i)T2 |
| 23 | 1+(4.89+2.82i)T+(11.5+19.9i)T2 |
| 29 | 1+(8.65+4.99i)T+(14.5+25.1i)T2 |
| 31 | 1+7.76T+31T2 |
| 37 | 1−1.31iT−37T2 |
| 41 | 1+(7.58−4.37i)T+(20.5−35.5i)T2 |
| 43 | 1+(−5.35+3.08i)T+(21.5−37.2i)T2 |
| 47 | 1+(−2.06−1.18i)T+(23.5+40.7i)T2 |
| 53 | 1+(−5.41−3.12i)T+(26.5+45.8i)T2 |
| 59 | 1+(3.28+5.68i)T+(−29.5+51.0i)T2 |
| 61 | 1+(−0.951+1.64i)T+(−30.5−52.8i)T2 |
| 67 | 1+(2.69−4.66i)T+(−33.5−58.0i)T2 |
| 71 | 1+(−2.60−4.51i)T+(−35.5+61.4i)T2 |
| 73 | 1+(−4.86−8.41i)T+(−36.5+63.2i)T2 |
| 79 | 1+(3.38+5.86i)T+(−39.5+68.4i)T2 |
| 83 | 1−1.55iT−83T2 |
| 89 | 1+(1.43+0.827i)T+(44.5+77.0i)T2 |
| 97 | 1+(9.10−5.25i)T+(48.5−84.0i)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
−9.967466987150960946165520191829, −9.308547908806763187052724063551, −8.346007894921600960307479326803, −7.49978211718722296178137038809, −6.69984287574709982469079500159, −5.93877142200240207183221045396, −5.34685217603021628380872250041, −4.04471717670539353682987546234, −2.48936096549231272333840592322, −1.87382408696336744318250762705,
0.04518693116869570399933857598, 1.73235003456435870438658007187, 3.60683015683843628922442457018, 4.09429918590598826620027157448, 5.07876236288811353795206020816, 5.78756404495192781481443000575, 6.97880678979906189343923337372, 7.60456931128306253457154587549, 8.825129557172126297379713103703, 9.474494470642116925182347578906