L(s) = 1 | + (1.41 − 0.0204i)2-s + (−1.50 + 0.869i)3-s + (1.99 − 0.0579i)4-s − 5-s + (−2.11 + 1.25i)6-s + (0.987 + 0.570i)7-s + (2.82 − 0.122i)8-s + (0.0109 − 0.0189i)9-s + (−1.41 + 0.0204i)10-s + (1.00 + 1.73i)11-s + (−2.95 + 1.82i)12-s + (0.482 + 3.57i)13-s + (1.40 + 0.785i)14-s + (1.50 − 0.869i)15-s + (3.99 − 0.231i)16-s + (1.50 − 2.59i)17-s + ⋯ |
L(s) = 1 | + (0.999 − 0.0144i)2-s + (−0.869 + 0.501i)3-s + (0.999 − 0.0289i)4-s − 0.447·5-s + (−0.861 + 0.514i)6-s + (0.373 + 0.215i)7-s + (0.999 − 0.0434i)8-s + (0.00365 − 0.00633i)9-s + (−0.447 + 0.00648i)10-s + (0.301 + 0.522i)11-s + (−0.854 + 0.526i)12-s + (0.133 + 0.990i)13-s + (0.376 + 0.210i)14-s + (0.388 − 0.224i)15-s + (0.998 − 0.0579i)16-s + (0.363 − 0.630i)17-s + ⋯ |
Λ(s)=(=(520s/2ΓC(s)L(s)(0.351−0.936i)Λ(2−s)
Λ(s)=(=(520s/2ΓC(s+1/2)L(s)(0.351−0.936i)Λ(1−s)
Degree: |
2 |
Conductor: |
520
= 23⋅5⋅13
|
Sign: |
0.351−0.936i
|
Analytic conductor: |
4.15222 |
Root analytic conductor: |
2.03769 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ520(101,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 520, ( :1/2), 0.351−0.936i)
|
Particular Values
L(1) |
≈ |
1.59259+1.10381i |
L(21) |
≈ |
1.59259+1.10381i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−1.41+0.0204i)T |
| 5 | 1+T |
| 13 | 1+(−0.482−3.57i)T |
good | 3 | 1+(1.50−0.869i)T+(1.5−2.59i)T2 |
| 7 | 1+(−0.987−0.570i)T+(3.5+6.06i)T2 |
| 11 | 1+(−1.00−1.73i)T+(−5.5+9.52i)T2 |
| 17 | 1+(−1.50+2.59i)T+(−8.5−14.7i)T2 |
| 19 | 1+(2.66−4.62i)T+(−9.5−16.4i)T2 |
| 23 | 1+(−1.81−3.13i)T+(−11.5+19.9i)T2 |
| 29 | 1+(−2.44+1.41i)T+(14.5−25.1i)T2 |
| 31 | 1−7.94iT−31T2 |
| 37 | 1+(4.33+7.51i)T+(−18.5+32.0i)T2 |
| 41 | 1+(−0.975+0.563i)T+(20.5−35.5i)T2 |
| 43 | 1+(−2.66−1.53i)T+(21.5+37.2i)T2 |
| 47 | 1−2.99iT−47T2 |
| 53 | 1+6.13iT−53T2 |
| 59 | 1+(−1.62+2.80i)T+(−29.5−51.0i)T2 |
| 61 | 1+(1.54+0.892i)T+(30.5+52.8i)T2 |
| 67 | 1+(7.15+12.4i)T+(−33.5+58.0i)T2 |
| 71 | 1+(−4.89−2.82i)T+(35.5+61.4i)T2 |
| 73 | 1+15.8iT−73T2 |
| 79 | 1−5.25T+79T2 |
| 83 | 1−12.2T+83T2 |
| 89 | 1+(1.76−1.01i)T+(44.5−77.0i)T2 |
| 97 | 1+(0.354+0.204i)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
−11.13806426092183192811025495136, −10.61178295639725319990275438190, −9.501178339743183115536191514299, −8.171387779773613522349872629257, −7.13776284564182424171928469434, −6.22710314253869711528069030697, −5.20547534146198696864356104170, −4.56100399679729024287730065246, −3.55106948883912533457875332637, −1.88850916033401435460695271758,
0.997444900436647094914910921975, 2.85856393840169570163738218745, 4.06170087812801651323987718697, 5.13458256638385869095384000735, 6.03665972830782231904516145036, 6.74732483164879632513670134760, 7.72708446560210740161771342888, 8.667598486195741759984972140689, 10.35146880405272444997197990605, 11.05278801846486837064384362293