L(s) = 1 | + (−2 − 3.46i)2-s + (−4.5 + 7.79i)3-s + (−7.99 + 13.8i)4-s + (30.5 + 52.8i)5-s + 36·6-s + 63.9·8-s + (−40.5 − 70.1i)9-s + (122. − 211. i)10-s + (18.2 − 31.6i)11-s + (−72 − 124. i)12-s + 34.5·13-s − 549.·15-s + (−128 − 221. i)16-s + (1.03e3 − 1.78e3i)17-s + (−162 + 280. i)18-s + (−226. − 391. i)19-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.546 + 0.946i)5-s + 0.408·6-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.386 − 0.669i)10-s + (0.0455 − 0.0788i)11-s + (−0.144 − 0.249i)12-s + 0.0567·13-s − 0.630·15-s + (−0.125 − 0.216i)16-s + (0.864 − 1.49i)17-s + (−0.117 + 0.204i)18-s + (−0.143 − 0.248i)19-s + ⋯ |
Λ(s)=(=(294s/2ΓC(s)L(s)(0.900+0.435i)Λ(6−s)
Λ(s)=(=(294s/2ΓC(s+5/2)L(s)(0.900+0.435i)Λ(1−s)
Degree: |
2 |
Conductor: |
294
= 2⋅3⋅72
|
Sign: |
0.900+0.435i
|
Analytic conductor: |
47.1528 |
Root analytic conductor: |
6.86679 |
Motivic weight: |
5 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ294(67,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 294, ( :5/2), 0.900+0.435i)
|
Particular Values
L(3) |
≈ |
1.549233644 |
L(21) |
≈ |
1.549233644 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(2+3.46i)T |
| 3 | 1+(4.5−7.79i)T |
| 7 | 1 |
good | 5 | 1+(−30.5−52.8i)T+(−1.56e3+2.70e3i)T2 |
| 11 | 1+(−18.2+31.6i)T+(−8.05e4−1.39e5i)T2 |
| 13 | 1−34.5T+3.71e5T2 |
| 17 | 1+(−1.03e3+1.78e3i)T+(−7.09e5−1.22e6i)T2 |
| 19 | 1+(226.+391.i)T+(−1.23e6+2.14e6i)T2 |
| 23 | 1+(842.+1.45e3i)T+(−3.21e6+5.57e6i)T2 |
| 29 | 1+4.76e3T+2.05e7T2 |
| 31 | 1+(−2.63e3+4.55e3i)T+(−1.43e7−2.47e7i)T2 |
| 37 | 1+(−6.41e3−1.11e4i)T+(−3.46e7+6.00e7i)T2 |
| 41 | 1+7.12e3T+1.15e8T2 |
| 43 | 1−1.11e4T+1.47e8T2 |
| 47 | 1+(−1.17e4−2.03e4i)T+(−1.14e8+1.98e8i)T2 |
| 53 | 1+(−3.51e3+6.08e3i)T+(−2.09e8−3.62e8i)T2 |
| 59 | 1+(−2.21e4+3.82e4i)T+(−3.57e8−6.19e8i)T2 |
| 61 | 1+(9.69e3+1.67e4i)T+(−4.22e8+7.31e8i)T2 |
| 67 | 1+(1.04e4−1.81e4i)T+(−6.75e8−1.16e9i)T2 |
| 71 | 1−7.98e4T+1.80e9T2 |
| 73 | 1+(−1.85e4+3.20e4i)T+(−1.03e9−1.79e9i)T2 |
| 79 | 1+(2.10e4+3.64e4i)T+(−1.53e9+2.66e9i)T2 |
| 83 | 1+6.31e3T+3.93e9T2 |
| 89 | 1+(−2.57e4−4.45e4i)T+(−2.79e9+4.83e9i)T2 |
| 97 | 1+1.27e5T+8.58e9T2 |
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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
−10.87923620233826606212448345331, −9.873576427387013293876501970022, −9.469139069364412359603211788551, −8.088016442433662616360030767258, −6.95685814746379592541523498874, −5.88214388309627707603379116910, −4.63544094189059521318662516656, −3.28212807909080380237813309844, −2.36027543222915141452321961214, −0.64728673517564259845963850735,
0.920144686825478246991350279564, 1.87309944060745736984388791615, 3.97246458168992858530276665448, 5.42929931438859363164514173476, 5.89305400142353704721752748212, 7.16303003550840066179163364602, 8.137161832933176238838386454617, 8.935348021224542482968687719946, 9.911902960021351977576981427562, 10.84510738049131759334532256222