L(s) = 1 | + (2 + 3.46i)2-s + (−4.5 + 7.79i)3-s + (−7.99 + 13.8i)4-s + (−51.7 − 89.6i)5-s − 36·6-s − 63.9·8-s + (−40.5 − 70.1i)9-s + (206. − 358. i)10-s + (−120. + 208. i)11-s + (−72 − 124. i)12-s + 805.·13-s + 931.·15-s + (−128 − 221. i)16-s + (646. − 1.12e3i)17-s + (162 − 280. i)18-s + (137. + 238. i)19-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.925 − 1.60i)5-s − 0.408·6-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.654 − 1.13i)10-s + (−0.299 + 0.518i)11-s + (−0.144 − 0.249i)12-s + 1.32·13-s + 1.06·15-s + (−0.125 − 0.216i)16-s + (0.542 − 0.939i)17-s + (0.117 − 0.204i)18-s + (0.0875 + 0.151i)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) |
≈ |
0.6746817880 |
L(21) |
≈ |
0.6746817880 |
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+(51.7+89.6i)T+(−1.56e3+2.70e3i)T2 |
| 11 | 1+(120.−208.i)T+(−8.05e4−1.39e5i)T2 |
| 13 | 1−805.T+3.71e5T2 |
| 17 | 1+(−646.+1.12e3i)T+(−7.09e5−1.22e6i)T2 |
| 19 | 1+(−137.−238.i)T+(−1.23e6+2.14e6i)T2 |
| 23 | 1+(1.89e3+3.28e3i)T+(−3.21e6+5.57e6i)T2 |
| 29 | 1−1.22e3T+2.05e7T2 |
| 31 | 1+(2.81e3−4.87e3i)T+(−1.43e7−2.47e7i)T2 |
| 37 | 1+(−4.53e3−7.86e3i)T+(−3.46e7+6.00e7i)T2 |
| 41 | 1+1.82e4T+1.15e8T2 |
| 43 | 1+1.17e4T+1.47e8T2 |
| 47 | 1+(−1.15e4−1.99e4i)T+(−1.14e8+1.98e8i)T2 |
| 53 | 1+(8.83e3−1.52e4i)T+(−2.09e8−3.62e8i)T2 |
| 59 | 1+(9.18e3−1.59e4i)T+(−3.57e8−6.19e8i)T2 |
| 61 | 1+(5.66e3+9.80e3i)T+(−4.22e8+7.31e8i)T2 |
| 67 | 1+(1.80e4−3.12e4i)T+(−6.75e8−1.16e9i)T2 |
| 71 | 1+6.34e4T+1.80e9T2 |
| 73 | 1+(−2.64e4+4.58e4i)T+(−1.03e9−1.79e9i)T2 |
| 79 | 1+(−2.42e4−4.20e4i)T+(−1.53e9+2.66e9i)T2 |
| 83 | 1−1.13e5T+3.93e9T2 |
| 89 | 1+(5.41e4+9.38e4i)T+(−2.79e9+4.83e9i)T2 |
| 97 | 1−9.96e4T+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
−11.67746599532900493366641035191, −10.36386178977784635372531329472, −9.129144071586550605333348675836, −8.456192258112928217952465554590, −7.63111350276851282839529306859, −6.23335823569125001920137255739, −5.05909528642798878902088862602, −4.50958013114299861744887360528, −3.44172828692050778258738652739, −1.09198641351113775959050559666,
0.19472807488905400749879755380, 1.83577275211195568490860382596, 3.28653292807623554903241064502, 3.81783796757596073178725100313, 5.69018738939597738078411138806, 6.46747154302176764150305047309, 7.58033851108019026727045520910, 8.407207801739574347314267956758, 10.05141424857939280733624960394, 10.86049903970582339848557020601