L(s) = 1 | + (−0.137 − 0.237i)2-s + (0.611 − 2.28i)3-s + (0.962 − 1.66i)4-s + (1.69 + 1.45i)5-s + (−0.627 + 0.168i)6-s + (0.334 + 0.193i)7-s − 1.07·8-s + (−2.23 − 1.29i)9-s + (0.112 − 0.604i)10-s + (4.21 + 1.12i)11-s + (−3.21 − 3.21i)12-s − 0.106i·14-s + (4.35 − 2.98i)15-s + (−1.77 − 3.07i)16-s + (−1.90 + 0.510i)17-s + 0.710i·18-s + ⋯ |
L(s) = 1 | + (−0.0971 − 0.168i)2-s + (0.353 − 1.31i)3-s + (0.481 − 0.833i)4-s + (0.759 + 0.650i)5-s + (−0.256 + 0.0686i)6-s + (0.126 + 0.0729i)7-s − 0.381·8-s + (−0.745 − 0.430i)9-s + (0.0356 − 0.191i)10-s + (1.27 + 0.340i)11-s + (−0.928 − 0.928i)12-s − 0.0283i·14-s + (1.12 − 0.771i)15-s + (−0.444 − 0.769i)16-s + (−0.462 + 0.123i)17-s + 0.167i·18-s + ⋯ |
Λ(s)=(=(845s/2ΓC(s)L(s)(−0.237+0.971i)Λ(2−s)
Λ(s)=(=(845s/2ΓC(s+1/2)L(s)(−0.237+0.971i)Λ(1−s)
Degree: |
2 |
Conductor: |
845
= 5⋅132
|
Sign: |
−0.237+0.971i
|
Analytic conductor: |
6.74735 |
Root analytic conductor: |
2.59756 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ845(258,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 845, ( :1/2), −0.237+0.971i)
|
Particular Values
L(1) |
≈ |
1.36235−1.73603i |
L(21) |
≈ |
1.36235−1.73603i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1+(−1.69−1.45i)T |
| 13 | 1 |
good | 2 | 1+(0.137+0.237i)T+(−1+1.73i)T2 |
| 3 | 1+(−0.611+2.28i)T+(−2.59−1.5i)T2 |
| 7 | 1+(−0.334−0.193i)T+(3.5+6.06i)T2 |
| 11 | 1+(−4.21−1.12i)T+(9.52+5.5i)T2 |
| 17 | 1+(1.90−0.510i)T+(14.7−8.5i)T2 |
| 19 | 1+(1.29+4.83i)T+(−16.4+9.5i)T2 |
| 23 | 1+(0.322+0.0863i)T+(19.9+11.5i)T2 |
| 29 | 1+(−7.07+4.08i)T+(14.5−25.1i)T2 |
| 31 | 1+(2.54+2.54i)T+31iT2 |
| 37 | 1+(4.17−2.41i)T+(18.5−32.0i)T2 |
| 41 | 1+(1.20−4.49i)T+(−35.5−20.5i)T2 |
| 43 | 1+(−1.76−6.58i)T+(−37.2+21.5i)T2 |
| 47 | 1−9.83iT−47T2 |
| 53 | 1+(7.17+7.17i)T+53iT2 |
| 59 | 1+(2.34−0.628i)T+(51.0−29.5i)T2 |
| 61 | 1+(5.32−9.22i)T+(−30.5−52.8i)T2 |
| 67 | 1+(−3.18−5.52i)T+(−33.5+58.0i)T2 |
| 71 | 1+(−4.20+1.12i)T+(61.4−35.5i)T2 |
| 73 | 1−6.08T+73T2 |
| 79 | 1−3.34iT−79T2 |
| 83 | 1−5.18iT−83T2 |
| 89 | 1+(−1.29+4.82i)T+(−77.0−44.5i)T2 |
| 97 | 1+(7.37−12.7i)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
−9.844261357530586674884388148844, −9.259665300066284837485172893167, −8.199104822976977027367351949936, −6.99581695931748720793388325363, −6.62395952954203989486898856464, −6.03703504765412887377169228081, −4.66156402332962929096616526744, −2.87318597791503193531854909952, −2.05441249413196440234872548205, −1.19853331229559434048470032974,
1.80793223431725146496982469510, 3.27743587114677822804621595826, 4.01262995199187825408598086235, 4.93796016354784491589561106930, 6.09413605633198623662852498594, 6.93193921853631595791713241633, 8.285088348182308208670433131168, 8.849054829090313782142440570722, 9.390187052216165872473830931763, 10.37138152539803138024712694751