L(s) = 1 | + (−8.52 − 14.7i)3-s + (40.1 − 28.1i)7-s + (−104. + 181. i)9-s + (−93.3 − 161. i)11-s + 200.·13-s + (−55.0 − 95.3i)17-s + (226. + 130. i)19-s + (−757. − 352. i)21-s + (−385. − 222. i)23-s + 2.18e3·27-s − 734.·29-s + (−406. + 234. i)31-s + (−1.59e3 + 2.75e3i)33-s + (−1.74e3 − 1.00e3i)37-s + (−1.71e3 − 2.96e3i)39-s + ⋯ |
L(s) = 1 | + (−0.946 − 1.64i)3-s + (0.818 − 0.574i)7-s + (−1.29 + 2.23i)9-s + (−0.771 − 1.33i)11-s + 1.18·13-s + (−0.190 − 0.329i)17-s + (0.626 + 0.361i)19-s + (−1.71 − 0.799i)21-s + (−0.727 − 0.420i)23-s + 3.00·27-s − 0.873·29-s + (−0.422 + 0.244i)31-s + (−1.46 + 2.52i)33-s + (−1.27 − 0.735i)37-s + (−1.12 − 1.94i)39-s + ⋯ |
Λ(s)=(=(700s/2ΓC(s)L(s)(0.481−0.876i)Λ(5−s)
Λ(s)=(=(700s/2ΓC(s+2)L(s)(0.481−0.876i)Λ(1−s)
Degree: |
2 |
Conductor: |
700
= 22⋅52⋅7
|
Sign: |
0.481−0.876i
|
Analytic conductor: |
72.3589 |
Root analytic conductor: |
8.50640 |
Motivic weight: |
4 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ700(649,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 700, ( :2), 0.481−0.876i)
|
Particular Values
L(25) |
≈ |
0.03873714477 |
L(21) |
≈ |
0.03873714477 |
L(3) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 5 | 1 |
| 7 | 1+(−40.1+28.1i)T |
good | 3 | 1+(8.52+14.7i)T+(−40.5+70.1i)T2 |
| 11 | 1+(93.3+161.i)T+(−7.32e3+1.26e4i)T2 |
| 13 | 1−200.T+2.85e4T2 |
| 17 | 1+(55.0+95.3i)T+(−4.17e4+7.23e4i)T2 |
| 19 | 1+(−226.−130.i)T+(6.51e4+1.12e5i)T2 |
| 23 | 1+(385.+222.i)T+(1.39e5+2.42e5i)T2 |
| 29 | 1+734.T+7.07e5T2 |
| 31 | 1+(406.−234.i)T+(4.61e5−7.99e5i)T2 |
| 37 | 1+(1.74e3+1.00e3i)T+(9.37e5+1.62e6i)T2 |
| 41 | 1+46.3iT−2.82e6T2 |
| 43 | 1−2.74e3iT−3.41e6T2 |
| 47 | 1+(2.14e3−3.71e3i)T+(−2.43e6−4.22e6i)T2 |
| 53 | 1+(2.85e3−1.64e3i)T+(3.94e6−6.83e6i)T2 |
| 59 | 1+(−207.+119.i)T+(6.05e6−1.04e7i)T2 |
| 61 | 1+(2.36e3+1.36e3i)T+(6.92e6+1.19e7i)T2 |
| 67 | 1+(−4.32e3+2.49e3i)T+(1.00e7−1.74e7i)T2 |
| 71 | 1+2.96e3T+2.54e7T2 |
| 73 | 1+(2.08e3+3.60e3i)T+(−1.41e7+2.45e7i)T2 |
| 79 | 1+(−1.76e3+3.05e3i)T+(−1.94e7−3.37e7i)T2 |
| 83 | 1+2.15e3T+4.74e7T2 |
| 89 | 1+(8.80e3+5.08e3i)T+(3.13e7+5.43e7i)T2 |
| 97 | 1−1.20e4T+8.85e7T2 |
show more | |
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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
−8.778658390587201294204727335781, −7.918924236319869074821856277381, −7.55026009034975205299663902481, −6.33476337397327589801005023508, −5.81267519886885297523542820227, −4.88155401814379669647413819182, −3.26837589391737576220336067425, −1.81465262854078303684712654760, −1.00315600308214764943368121749, −0.01188622227613067052854266146,
1.81957876509016754811152417419, 3.44749953374816190913432521748, 4.33937054026669557792651656144, 5.23756277113857683739319712095, 5.65450397183183544988223500394, 6.91540341942404823271912165411, 8.243021402107667867685429430687, 9.054937135511830020222735769340, 9.923974898270311712392853141870, 10.50339078694995105223684667581