L(s) = 1 | + (−1.56 − 0.746i)3-s + (0.842 − 1.45i)5-s + (1.88 + 2.33i)9-s + (−3.38 + 1.95i)11-s + (5.24 + 3.02i)13-s + (−2.40 + 1.65i)15-s − 0.402·17-s − 0.168i·19-s + (−7.69 − 4.44i)23-s + (1.07 + 1.86i)25-s + (−1.20 − 5.05i)27-s + (−6.15 + 3.55i)29-s + (−5.44 − 3.14i)31-s + (6.74 − 0.525i)33-s − 6.26·37-s + ⋯ |
L(s) = 1 | + (−0.902 − 0.431i)3-s + (0.376 − 0.652i)5-s + (0.628 + 0.778i)9-s + (−1.01 + 0.588i)11-s + (1.45 + 0.839i)13-s + (−0.621 + 0.426i)15-s − 0.0976·17-s − 0.0385i·19-s + (−1.60 − 0.926i)23-s + (0.215 + 0.373i)25-s + (−0.230 − 0.972i)27-s + (−1.14 + 0.659i)29-s + (−0.977 − 0.564i)31-s + (1.17 − 0.0913i)33-s − 1.02·37-s + ⋯ |
Λ(s)=(=(1764s/2ΓC(s)L(s)(−0.139−0.990i)Λ(2−s)
Λ(s)=(=(1764s/2ΓC(s+1/2)L(s)(−0.139−0.990i)Λ(1−s)
Degree: |
2 |
Conductor: |
1764
= 22⋅32⋅72
|
Sign: |
−0.139−0.990i
|
Analytic conductor: |
14.0856 |
Root analytic conductor: |
3.75308 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1764(293,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1764, ( :1/2), −0.139−0.990i)
|
Particular Values
L(1) |
≈ |
0.5847496569 |
L(21) |
≈ |
0.5847496569 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1+(1.56+0.746i)T |
| 7 | 1 |
good | 5 | 1+(−0.842+1.45i)T+(−2.5−4.33i)T2 |
| 11 | 1+(3.38−1.95i)T+(5.5−9.52i)T2 |
| 13 | 1+(−5.24−3.02i)T+(6.5+11.2i)T2 |
| 17 | 1+0.402T+17T2 |
| 19 | 1+0.168iT−19T2 |
| 23 | 1+(7.69+4.44i)T+(11.5+19.9i)T2 |
| 29 | 1+(6.15−3.55i)T+(14.5−25.1i)T2 |
| 31 | 1+(5.44+3.14i)T+(15.5+26.8i)T2 |
| 37 | 1+6.26T+37T2 |
| 41 | 1+(1.64−2.85i)T+(−20.5−35.5i)T2 |
| 43 | 1+(−1.80−3.12i)T+(−21.5+37.2i)T2 |
| 47 | 1+(−4.38−7.59i)T+(−23.5+40.7i)T2 |
| 53 | 1−5.71iT−53T2 |
| 59 | 1+(−2.25+3.89i)T+(−29.5−51.0i)T2 |
| 61 | 1+(4.43−2.56i)T+(30.5−52.8i)T2 |
| 67 | 1+(−2.95+5.11i)T+(−33.5−58.0i)T2 |
| 71 | 1−11.4iT−71T2 |
| 73 | 1−6.99iT−73T2 |
| 79 | 1+(0.603+1.04i)T+(−39.5+68.4i)T2 |
| 83 | 1+(0.181+0.314i)T+(−41.5+71.8i)T2 |
| 89 | 1+2.77T+89T2 |
| 97 | 1+(−0.508+0.293i)T+(48.5−84.0i)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
−9.513769053738589601788038837930, −8.719065218304145376795696318804, −7.86535037515871522211910498126, −7.07958847538085120645929697908, −6.14829377286220336089982846430, −5.58936384275703769644907796702, −4.72664061944152547350451080646, −3.87984080483920390756244554746, −2.18800663183378830740874382766, −1.36320412147329382305526678779,
0.24818145278534970541957540140, 1.88176204295001027192528452724, 3.32522407753686649790336347278, 3.91576742243472489532784821206, 5.40785595974211491049284878211, 5.68168139779484294659644434784, 6.46053608091570583754817275334, 7.43525341924736947118971213510, 8.290199366196608885530001488438, 9.180444055983000122733650508153