L(s) = 1 | + (−3.40 − 1.10i)2-s + (−3.07 + 2.23i)3-s + (7.11 + 5.16i)4-s + (−0.690 − 2.12i)5-s + (12.9 − 4.19i)6-s + (4.78 − 6.58i)7-s + (−10.0 − 13.8i)8-s + (1.67 − 5.15i)9-s + 7.99i·10-s + (7.96 − 7.58i)11-s − 33.3·12-s + (20.9 + 6.79i)13-s + (−23.5 + 17.1i)14-s + (6.87 + 4.99i)15-s + (8.07 + 24.8i)16-s + (0.261 − 0.0849i)17-s + ⋯ |
L(s) = 1 | + (−1.70 − 0.552i)2-s + (−1.02 + 0.744i)3-s + (1.77 + 1.29i)4-s + (−0.138 − 0.425i)5-s + (2.15 − 0.699i)6-s + (0.683 − 0.940i)7-s + (−1.25 − 1.73i)8-s + (0.186 − 0.573i)9-s + 0.799i·10-s + (0.724 − 0.689i)11-s − 2.78·12-s + (1.60 + 0.522i)13-s + (−1.68 + 1.22i)14-s + (0.458 + 0.332i)15-s + (0.504 + 1.55i)16-s + (0.0153 − 0.00499i)17-s + ⋯ |
Λ(s)=(=(55s/2ΓC(s)L(s)(0.673+0.739i)Λ(3−s)
Λ(s)=(=(55s/2ΓC(s+1)L(s)(0.673+0.739i)Λ(1−s)
Degree: |
2 |
Conductor: |
55
= 5⋅11
|
Sign: |
0.673+0.739i
|
Analytic conductor: |
1.49864 |
Root analytic conductor: |
1.22419 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ55(46,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 55, ( :1), 0.673+0.739i)
|
Particular Values
L(23) |
≈ |
0.388122−0.171476i |
L(21) |
≈ |
0.388122−0.171476i |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1+(0.690+2.12i)T |
| 11 | 1+(−7.96+7.58i)T |
good | 2 | 1+(3.40+1.10i)T+(3.23+2.35i)T2 |
| 3 | 1+(3.07−2.23i)T+(2.78−8.55i)T2 |
| 7 | 1+(−4.78+6.58i)T+(−15.1−46.6i)T2 |
| 13 | 1+(−20.9−6.79i)T+(136.+99.3i)T2 |
| 17 | 1+(−0.261+0.0849i)T+(233.−169.i)T2 |
| 19 | 1+(9.69+13.3i)T+(−111.+343.i)T2 |
| 23 | 1−1.44T+529T2 |
| 29 | 1+(−24.5+33.7i)T+(−259.−799.i)T2 |
| 31 | 1+(−0.152+0.468i)T+(−777.−564.i)T2 |
| 37 | 1+(−17.6−12.8i)T+(423.+1.30e3i)T2 |
| 41 | 1+(−29.5−40.6i)T+(−519.+1.59e3i)T2 |
| 43 | 1+62.2iT−1.84e3T2 |
| 47 | 1+(10.9−7.94i)T+(682.−2.10e3i)T2 |
| 53 | 1+(−1.14+3.52i)T+(−2.27e3−1.65e3i)T2 |
| 59 | 1+(−7.28−5.29i)T+(1.07e3+3.31e3i)T2 |
| 61 | 1+(78.3−25.4i)T+(3.01e3−2.18e3i)T2 |
| 67 | 1+36.5T+4.48e3T2 |
| 71 | 1+(−29.6−91.1i)T+(−4.07e3+2.96e3i)T2 |
| 73 | 1+(−54.0+74.3i)T+(−1.64e3−5.06e3i)T2 |
| 79 | 1+(32.6+10.6i)T+(5.04e3+3.66e3i)T2 |
| 83 | 1+(16.9−5.50i)T+(5.57e3−4.04e3i)T2 |
| 89 | 1−24.9T+7.92e3T2 |
| 97 | 1+(29.0−89.3i)T+(−7.61e3−5.53e3i)T2 |
show more | |
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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
−15.69810703907578137543848776165, −13.66894116987764217937985084180, −11.71560414705074123990276664077, −11.16205385078750279463003328156, −10.44823560711641212058195469383, −9.109331832553966072694825695874, −8.088615656008583006135202029825, −6.38893402101444723127795765848, −4.19768308509162719413946072869, −0.965149467627808304917763812519,
1.45601656647546670670135299283, 5.85631338385764292429693020540, 6.67457219709163743986137057019, 7.985209103822211211302269407054, 9.038989031574201683906556744726, 10.63931575582013890082092095722, 11.40185817597018250635761183582, 12.47074995199870384239491982284, 14.60538415604074023142527429508, 15.59219435584751185382630967461