L(s) = 1 | + (−0.618 − 1.90i)2-s + (−6.71 + 4.88i)3-s + (−3.23 + 2.35i)4-s + (13.4 + 9.76i)6-s + 11.9·7-s + (6.47 + 4.70i)8-s + (12.9 − 39.9i)9-s + (16.7 + 51.6i)11-s + (10.2 − 31.5i)12-s + (20.9 − 64.4i)13-s + (−7.39 − 22.7i)14-s + (4.94 − 15.2i)16-s + (−8.59 − 6.24i)17-s − 83.9·18-s + (−18.5 − 13.5i)19-s + ⋯ |
L(s) = 1 | + (−0.218 − 0.672i)2-s + (−1.29 + 0.939i)3-s + (−0.404 + 0.293i)4-s + (0.914 + 0.664i)6-s + 0.645·7-s + (0.286 + 0.207i)8-s + (0.480 − 1.47i)9-s + (0.460 + 1.41i)11-s + (0.246 − 0.760i)12-s + (0.447 − 1.37i)13-s + (−0.141 − 0.434i)14-s + (0.0772 − 0.237i)16-s + (−0.122 − 0.0891i)17-s − 1.09·18-s + (−0.224 − 0.163i)19-s + ⋯ |
Λ(s)=(=(250s/2ΓC(s)L(s)(−0.455−0.890i)Λ(4−s)
Λ(s)=(=(250s/2ΓC(s+3/2)L(s)(−0.455−0.890i)Λ(1−s)
Degree: |
2 |
Conductor: |
250
= 2⋅53
|
Sign: |
−0.455−0.890i
|
Analytic conductor: |
14.7504 |
Root analytic conductor: |
3.84063 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ250(151,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 250, ( :3/2), −0.455−0.890i)
|
Particular Values
L(2) |
≈ |
0.316107+0.517109i |
L(21) |
≈ |
0.316107+0.517109i |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(0.618+1.90i)T |
| 5 | 1 |
good | 3 | 1+(6.71−4.88i)T+(8.34−25.6i)T2 |
| 7 | 1−11.9T+343T2 |
| 11 | 1+(−16.7−51.6i)T+(−1.07e3+782.i)T2 |
| 13 | 1+(−20.9+64.4i)T+(−1.77e3−1.29e3i)T2 |
| 17 | 1+(8.59+6.24i)T+(1.51e3+4.67e3i)T2 |
| 19 | 1+(18.5+13.5i)T+(2.11e3+6.52e3i)T2 |
| 23 | 1+(−45.4−140.i)T+(−9.84e3+7.15e3i)T2 |
| 29 | 1+(166.−121.i)T+(7.53e3−2.31e4i)T2 |
| 31 | 1+(−70.9−51.5i)T+(9.20e3+2.83e4i)T2 |
| 37 | 1+(127.−393.i)T+(−4.09e4−2.97e4i)T2 |
| 41 | 1+(20.9−64.3i)T+(−5.57e4−4.05e4i)T2 |
| 43 | 1+471.T+7.95e4T2 |
| 47 | 1+(−53.4+38.8i)T+(3.20e4−9.87e4i)T2 |
| 53 | 1+(−255.+185.i)T+(4.60e4−1.41e5i)T2 |
| 59 | 1+(31.3−96.3i)T+(−1.66e5−1.20e5i)T2 |
| 61 | 1+(125.+385.i)T+(−1.83e5+1.33e5i)T2 |
| 67 | 1+(425.+308.i)T+(9.29e4+2.86e5i)T2 |
| 71 | 1+(−386.+280.i)T+(1.10e5−3.40e5i)T2 |
| 73 | 1+(−264.−815.i)T+(−3.14e5+2.28e5i)T2 |
| 79 | 1+(444.−322.i)T+(1.52e5−4.68e5i)T2 |
| 83 | 1+(−232.−169.i)T+(1.76e5+5.43e5i)T2 |
| 89 | 1+(12.3+37.9i)T+(−5.70e5+4.14e5i)T2 |
| 97 | 1+(833.−605.i)T+(2.82e5−8.68e5i)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
−11.65881636332235965650743175777, −10.99260727311940229972259452559, −10.17077617291388288120641946119, −9.520478843970382410709517155349, −8.182649435778756152928366386462, −6.84099289432263308206392499347, −5.35727673889344020993095979882, −4.75904459517460167934848188134, −3.50474614082634548632521361415, −1.42015569879401814130074604148,
0.33819275537822039371571172382, 1.65118809403838743887997952946, 4.22043468190699007724740408254, 5.53480154067957260945344033925, 6.29917445547807911518291997972, 7.02392613189338579829953290987, 8.216812338144935410800777692422, 9.074172134806216855438541010358, 10.72216828499353197186171295261, 11.33176309870465686291007345521