L(s) = 1 | + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.978 − 0.207i)5-s + (−0.809 − 0.587i)8-s + (0.5 + 0.866i)10-s + (0.978 + 0.207i)13-s + (0.309 − 0.951i)16-s + (0.978 − 0.207i)17-s + (−0.913 + 0.406i)19-s + (−0.669 + 0.743i)20-s + (−0.5 − 0.866i)23-s + (0.913 − 0.406i)25-s + (0.104 + 0.994i)26-s + (0.913 + 0.406i)29-s + (−0.309 − 0.951i)31-s + 32-s + ⋯ |
L(s) = 1 | + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.978 − 0.207i)5-s + (−0.809 − 0.587i)8-s + (0.5 + 0.866i)10-s + (0.978 + 0.207i)13-s + (0.309 − 0.951i)16-s + (0.978 − 0.207i)17-s + (−0.913 + 0.406i)19-s + (−0.669 + 0.743i)20-s + (−0.5 − 0.866i)23-s + (0.913 − 0.406i)25-s + (0.104 + 0.994i)26-s + (0.913 + 0.406i)29-s + (−0.309 − 0.951i)31-s + 32-s + ⋯ |
Λ(s)=(=(693s/2ΓR(s+1)L(s)(0.758+0.651i)Λ(1−s)
Λ(s)=(=(693s/2ΓR(s+1)L(s)(0.758+0.651i)Λ(1−s)
Degree: |
1 |
Conductor: |
693
= 32⋅7⋅11
|
Sign: |
0.758+0.651i
|
Analytic conductor: |
74.4731 |
Root analytic conductor: |
74.4731 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ693(367,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 693, (1: ), 0.758+0.651i)
|
Particular Values
L(21) |
≈ |
2.647729067+0.9800659207i |
L(21) |
≈ |
2.647729067+0.9800659207i |
L(1) |
≈ |
1.334937212+0.5639099594i |
L(1) |
≈ |
1.334937212+0.5639099594i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 7 | 1 |
| 11 | 1 |
good | 2 | 1+(0.309+0.951i)T |
| 5 | 1+(0.978−0.207i)T |
| 13 | 1+(0.978+0.207i)T |
| 17 | 1+(0.978−0.207i)T |
| 19 | 1+(−0.913+0.406i)T |
| 23 | 1+(−0.5−0.866i)T |
| 29 | 1+(0.913+0.406i)T |
| 31 | 1+(−0.309−0.951i)T |
| 37 | 1+(−0.104−0.994i)T |
| 41 | 1+(−0.913+0.406i)T |
| 43 | 1+(−0.5−0.866i)T |
| 47 | 1+(0.809+0.587i)T |
| 53 | 1+(0.669−0.743i)T |
| 59 | 1+(0.809−0.587i)T |
| 61 | 1+(−0.309+0.951i)T |
| 67 | 1+T |
| 71 | 1+(0.309−0.951i)T |
| 73 | 1+(−0.913−0.406i)T |
| 79 | 1+(0.309+0.951i)T |
| 83 | 1+(0.978−0.207i)T |
| 89 | 1+(0.5−0.866i)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
−22.04039510037612925610246208868, −21.55538062804224888114662277182, −20.90697776900145259978762159100, −20.088237789633366573212038905577, −19.12901034438570451469583280199, −18.41417197449958583692808276776, −17.68312535028254049177367520012, −16.91723330498582828379987274298, −15.60899641662050974682149584741, −14.68697472533284021880336236639, −13.82248004118884667867185964649, −13.32232427087317948141201932704, −12.4076918579399594971393917648, −11.49563983435750778422194749485, −10.48336148137959934485126609184, −10.06677411787545066420634661022, −9.01470398695150165345704795496, −8.234008439089132777789929367077, −6.6421709364802265419029198681, −5.81853886279077279625331095311, −5.021656034949655206464358140188, −3.79254528317565054754319989948, −2.926384920775196442622041013074, −1.8491516038531211668326792275, −0.9933688035214323105365494811,
0.70885403910478471791383439893, 2.11371692983017508510921620168, 3.45635513008488608000740339055, 4.44836032228134808395616059152, 5.50879420604035225836239524127, 6.13616471712851348909019139089, 6.94090174652337558036055728671, 8.1892967990504261946049180714, 8.78721651460016447267275461576, 9.778057425486343903808144139055, 10.609098756283150988391773371804, 12.04870227631058172939694991982, 12.81392706716919068418147583855, 13.63594571178407498672258166697, 14.27421692165334931386460794817, 15.03859805947609692031958017768, 16.23003162111020013572450126831, 16.62793702120006656659375054770, 17.53244301173236362778267480610, 18.31587410490430778563213364019, 18.933689580243507959384067270, 20.46934135586460068666501962809, 21.12079809908748784180332458472, 21.7993287954622472547409214529, 22.67952522537950082598894583660