L(s) = 1 | + (−0.406 + 0.913i)2-s + (−0.669 − 0.743i)4-s + (0.866 − 0.5i)7-s + (0.951 − 0.309i)8-s + (0.913 + 0.406i)11-s + (−0.406 − 0.913i)13-s + (0.104 + 0.994i)14-s + (−0.104 + 0.994i)16-s + (−0.951 + 0.309i)17-s + (−0.309 − 0.951i)19-s + (−0.743 + 0.669i)22-s + (−0.994 + 0.104i)23-s + 26-s + (−0.951 − 0.309i)28-s + (0.978 − 0.207i)29-s + ⋯ |
L(s) = 1 | + (−0.406 + 0.913i)2-s + (−0.669 − 0.743i)4-s + (0.866 − 0.5i)7-s + (0.951 − 0.309i)8-s + (0.913 + 0.406i)11-s + (−0.406 − 0.913i)13-s + (0.104 + 0.994i)14-s + (−0.104 + 0.994i)16-s + (−0.951 + 0.309i)17-s + (−0.309 − 0.951i)19-s + (−0.743 + 0.669i)22-s + (−0.994 + 0.104i)23-s + 26-s + (−0.951 − 0.309i)28-s + (0.978 − 0.207i)29-s + ⋯ |
Λ(s)=(=(225s/2ΓR(s+1)L(s)(0.663−0.747i)Λ(1−s)
Λ(s)=(=(225s/2ΓR(s+1)L(s)(0.663−0.747i)Λ(1−s)
Degree: |
1 |
Conductor: |
225
= 32⋅52
|
Sign: |
0.663−0.747i
|
Analytic conductor: |
24.1796 |
Root analytic conductor: |
24.1796 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ225(142,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 225, (1: ), 0.663−0.747i)
|
Particular Values
L(21) |
≈ |
1.034821875−0.4650673910i |
L(21) |
≈ |
1.034821875−0.4650673910i |
L(1) |
≈ |
0.8504837098+0.1145423349i |
L(1) |
≈ |
0.8504837098+0.1145423349i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 5 | 1 |
good | 2 | 1+(−0.406+0.913i)T |
| 7 | 1+(0.866−0.5i)T |
| 11 | 1+(0.913+0.406i)T |
| 13 | 1+(−0.406−0.913i)T |
| 17 | 1+(−0.951+0.309i)T |
| 19 | 1+(−0.309−0.951i)T |
| 23 | 1+(−0.994+0.104i)T |
| 29 | 1+(0.978−0.207i)T |
| 31 | 1+(−0.978−0.207i)T |
| 37 | 1+(0.587−0.809i)T |
| 41 | 1+(0.913−0.406i)T |
| 43 | 1+(−0.866+0.5i)T |
| 47 | 1+(−0.207−0.978i)T |
| 53 | 1+(−0.951−0.309i)T |
| 59 | 1+(−0.913+0.406i)T |
| 61 | 1+(0.913+0.406i)T |
| 67 | 1+(0.207−0.978i)T |
| 71 | 1+(0.309−0.951i)T |
| 73 | 1+(0.587+0.809i)T |
| 79 | 1+(0.978−0.207i)T |
| 83 | 1+(−0.743−0.669i)T |
| 89 | 1+(0.809−0.587i)T |
| 97 | 1+(−0.207−0.978i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−26.7665856670483010896240402073, −25.482342147762743772365619194559, −24.54679031446176032109844645585, −23.500528707698973947141159658338, −22.0778134132227559936528525909, −21.75737379589625788652291729718, −20.655179297211885429191177220773, −19.78316832834842964734073202562, −18.84303801964349970552666949144, −18.016179213422404793290389028632, −17.11288587233936586538361840335, −16.1436221995463493452145687694, −14.53807227226687646167457402935, −13.92056837738616302391596784222, −12.48851545280139254412516869479, −11.70376136653097890726975304694, −10.98674805496008100956219197314, −9.66806074778819810957591653418, −8.80061780635607203910250765654, −7.91753520986959132403514673094, −6.43940277059398070582110099253, −4.82804118111253549739943806952, −3.89910494184633858581888173923, −2.3620396002793992578402198503, −1.38910687953855340356959438536,
0.45203202733925016807946534625, 1.910656669816904722079151298626, 4.0847019026501770664409082102, 4.95830533543306343270708693171, 6.271409576259364786249997886, 7.28537421126516292618731398293, 8.20012794699823255582506761208, 9.22478934725356535074884082227, 10.33936639790869829252599790042, 11.31779833208518226732273975074, 12.81011057356473898429584693875, 13.92464445174741852914269117340, 14.76269294691061953192599306314, 15.53136578063990939932531006322, 16.75994520850473732550392115043, 17.641104829730485752305693101015, 18.01866571888775662247367174706, 19.70998497490319326695987828371, 19.99204662244164352324611256360, 21.653137509023075548647021975243, 22.53226581383766858907726262359, 23.54940213539606868201842633347, 24.35815944315603882563513026397, 25.05858495085864845339610289099, 26.1000804579136220995965624056