L(s) = 1 | + (−0.539 + 1.30i)2-s + (−0.149 + 0.360i)3-s + (−1.41 − 1.40i)4-s + (2.11 − 0.876i)5-s + (−0.390 − 0.389i)6-s + (0.707 − 0.707i)7-s + (2.60 − 1.09i)8-s + (2.01 + 2.01i)9-s + (0.00537 + 3.23i)10-s + (−0.619 − 1.49i)11-s + (0.719 − 0.300i)12-s + (2.00 + 0.829i)13-s + (0.543 + 1.30i)14-s + 0.892i·15-s + (0.0265 + 3.99i)16-s + 2.15i·17-s + ⋯ |
L(s) = 1 | + (−0.381 + 0.924i)2-s + (−0.0861 + 0.208i)3-s + (−0.709 − 0.704i)4-s + (0.945 − 0.391i)5-s + (−0.159 − 0.158i)6-s + (0.267 − 0.267i)7-s + (0.921 − 0.387i)8-s + (0.671 + 0.671i)9-s + (0.00169 + 1.02i)10-s + (−0.186 − 0.450i)11-s + (0.207 − 0.0868i)12-s + (0.555 + 0.230i)13-s + (0.145 + 0.348i)14-s + 0.230i·15-s + (0.00663 + 0.999i)16-s + 0.522i·17-s + ⋯ |
Λ(s)=(=(224s/2ΓC(s)L(s)(0.548−0.836i)Λ(2−s)
Λ(s)=(=(224s/2ΓC(s+1/2)L(s)(0.548−0.836i)Λ(1−s)
Degree: |
2 |
Conductor: |
224
= 25⋅7
|
Sign: |
0.548−0.836i
|
Analytic conductor: |
1.78864 |
Root analytic conductor: |
1.33740 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ224(141,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 224, ( :1/2), 0.548−0.836i)
|
Particular Values
L(1) |
≈ |
1.00661+0.543429i |
L(21) |
≈ |
1.00661+0.543429i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(0.539−1.30i)T |
| 7 | 1+(−0.707+0.707i)T |
good | 3 | 1+(0.149−0.360i)T+(−2.12−2.12i)T2 |
| 5 | 1+(−2.11+0.876i)T+(3.53−3.53i)T2 |
| 11 | 1+(0.619+1.49i)T+(−7.77+7.77i)T2 |
| 13 | 1+(−2.00−0.829i)T+(9.19+9.19i)T2 |
| 17 | 1−2.15iT−17T2 |
| 19 | 1+(−1.09−0.453i)T+(13.4+13.4i)T2 |
| 23 | 1+(1.44+1.44i)T+23iT2 |
| 29 | 1+(0.483−1.16i)T+(−20.5−20.5i)T2 |
| 31 | 1+0.217T+31T2 |
| 37 | 1+(−9.82+4.06i)T+(26.1−26.1i)T2 |
| 41 | 1+(4.75+4.75i)T+41iT2 |
| 43 | 1+(1.88+4.55i)T+(−30.4+30.4i)T2 |
| 47 | 1−6.77iT−47T2 |
| 53 | 1+(2.22+5.37i)T+(−37.4+37.4i)T2 |
| 59 | 1+(6.35−2.63i)T+(41.7−41.7i)T2 |
| 61 | 1+(1.20−2.90i)T+(−43.1−43.1i)T2 |
| 67 | 1+(−5.29+12.7i)T+(−47.3−47.3i)T2 |
| 71 | 1+(7.30−7.30i)T−71iT2 |
| 73 | 1+(9.12+9.12i)T+73iT2 |
| 79 | 1−5.17iT−79T2 |
| 83 | 1+(15.2+6.29i)T+(58.6+58.6i)T2 |
| 89 | 1+(9.77−9.77i)T−89iT2 |
| 97 | 1+12.5T+97T2 |
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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
−12.80629526840971724977687640398, −11.05890701979596525204847750229, −10.26018046201443735966098569740, −9.420297283834945834971373608728, −8.425230778718341434075129686948, −7.45383361524184311350374965287, −6.18601899253058517847373247085, −5.33687160352878086027549000896, −4.20541938369134733408710340579, −1.60056577421926252355573147393,
1.52917719245709547359565113296, 2.90375194813822476755865130640, 4.44164614841454479882458164957, 5.89132953206478689195814449567, 7.16115690482810726282917003991, 8.341752092609066090678925535953, 9.634096405998284376860499310681, 9.939959015562660156169598837960, 11.17121777899031712003885337694, 11.98566440572076954420861961267