L(s) = 1 | + (0.5 + 0.866i)7-s + (0.104 − 0.994i)11-s + (0.104 + 0.994i)13-s + (−0.309 + 0.951i)17-s + (0.309 − 0.951i)19-s + (0.913 − 0.406i)23-s + (0.669 − 0.743i)29-s + (−0.669 − 0.743i)31-s + (0.809 − 0.587i)37-s + (0.104 + 0.994i)41-s + (−0.5 − 0.866i)43-s + (0.669 − 0.743i)47-s + (−0.5 + 0.866i)49-s + (0.309 + 0.951i)53-s + (0.104 + 0.994i)59-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)7-s + (0.104 − 0.994i)11-s + (0.104 + 0.994i)13-s + (−0.309 + 0.951i)17-s + (0.309 − 0.951i)19-s + (0.913 − 0.406i)23-s + (0.669 − 0.743i)29-s + (−0.669 − 0.743i)31-s + (0.809 − 0.587i)37-s + (0.104 + 0.994i)41-s + (−0.5 − 0.866i)43-s + (0.669 − 0.743i)47-s + (−0.5 + 0.866i)49-s + (0.309 + 0.951i)53-s + (0.104 + 0.994i)59-s + ⋯ |
Λ(s)=(=(1800s/2ΓR(s)L(s)(0.959+0.282i)Λ(1−s)
Λ(s)=(=(1800s/2ΓR(s)L(s)(0.959+0.282i)Λ(1−s)
Degree: |
1 |
Conductor: |
1800
= 23⋅32⋅52
|
Sign: |
0.959+0.282i
|
Analytic conductor: |
8.35916 |
Root analytic conductor: |
8.35916 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1800(371,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 1800, (0: ), 0.959+0.282i)
|
Particular Values
L(21) |
≈ |
1.759554714+0.2535557077i |
L(21) |
≈ |
1.759554714+0.2535557077i |
L(1) |
≈ |
1.188166106+0.08678488697i |
L(1) |
≈ |
1.188166106+0.08678488697i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
| 5 | 1 |
good | 7 | 1+(0.5+0.866i)T |
| 11 | 1+(0.104−0.994i)T |
| 13 | 1+(0.104+0.994i)T |
| 17 | 1+(−0.309+0.951i)T |
| 19 | 1+(0.309−0.951i)T |
| 23 | 1+(0.913−0.406i)T |
| 29 | 1+(0.669−0.743i)T |
| 31 | 1+(−0.669−0.743i)T |
| 37 | 1+(0.809−0.587i)T |
| 41 | 1+(0.104+0.994i)T |
| 43 | 1+(−0.5−0.866i)T |
| 47 | 1+(0.669−0.743i)T |
| 53 | 1+(0.309+0.951i)T |
| 59 | 1+(0.104+0.994i)T |
| 61 | 1+(0.104−0.994i)T |
| 67 | 1+(0.669+0.743i)T |
| 71 | 1+(0.309+0.951i)T |
| 73 | 1+(−0.809−0.587i)T |
| 79 | 1+(−0.669+0.743i)T |
| 83 | 1+(0.978−0.207i)T |
| 89 | 1+(0.809+0.587i)T |
| 97 | 1+(0.669−0.743i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−20.31769681038171396966522477742, −19.63228149013610155346792794234, −18.54511012409298378239593900083, −17.82762939582480868896357115450, −17.41355790218552314228706570650, −16.47012098017569267276192564726, −15.78984370284282948874010780349, −14.84737239131098550973864812272, −14.346621673733952080000049888344, −13.45469682459760165317762953357, −12.763207316503517087572049160907, −11.9635045194877071584248823706, −11.086966304344001737427410157881, −10.3916020210057211072421999333, −9.7254605226036977272766916812, −8.80118947914087871158539087709, −7.78651805528252078913332254640, −7.32124056867254639677471705063, −6.49215028709643826843001239178, −5.24786259301647408530786192560, −4.78530589596228940959797807513, −3.742684991896335812762490428922, −2.92025873747630888290018918719, −1.72527547100092491031411791742, −0.8606745948746860729536106508,
0.900046445456895273829818705892, 2.05505737891670183408208720571, 2.78085501189164093919793187807, 3.912925034109424307491870966688, 4.70803208451950802127820670806, 5.67157625639123768528517084702, 6.30727630142096983330828362950, 7.21574379180619511055484701467, 8.29786085813350075828122747178, 8.81917902840999379145999500344, 9.45420418849279789757972699321, 10.65675314224221897812069779264, 11.34613395321964245360001906527, 11.8104293117783363618917352095, 12.85711063425353387818241718900, 13.53173451855245171645321450297, 14.36496206599711199841074416063, 15.074419216899465376146227454201, 15.724550296944190343258134175762, 16.63704298118605262496239921615, 17.20513241536625156009614855492, 18.15868971930268820774893091612, 18.78565049047088805673571463087, 19.34249614256485559089181632816, 20.20489716999850082197217458024