L(s) = 1 | + (−0.826 + 0.563i)3-s + (−0.733 + 0.680i)5-s + (0.365 − 0.930i)9-s + (−0.365 − 0.930i)11-s + (0.623 − 0.781i)13-s + (0.222 − 0.974i)15-s + (−0.5 + 0.866i)17-s + (−0.826 − 0.563i)19-s + (−0.955 + 0.294i)23-s + (0.0747 − 0.997i)25-s + (0.222 + 0.974i)27-s + (−0.955 − 0.294i)31-s + (0.826 + 0.563i)33-s + (0.365 − 0.930i)37-s + (−0.0747 + 0.997i)39-s + ⋯ |
L(s) = 1 | + (−0.826 + 0.563i)3-s + (−0.733 + 0.680i)5-s + (0.365 − 0.930i)9-s + (−0.365 − 0.930i)11-s + (0.623 − 0.781i)13-s + (0.222 − 0.974i)15-s + (−0.5 + 0.866i)17-s + (−0.826 − 0.563i)19-s + (−0.955 + 0.294i)23-s + (0.0747 − 0.997i)25-s + (0.222 + 0.974i)27-s + (−0.955 − 0.294i)31-s + (0.826 + 0.563i)33-s + (0.365 − 0.930i)37-s + (−0.0747 + 0.997i)39-s + ⋯ |
Λ(s)=(=(812s/2ΓR(s+1)L(s)(−0.604+0.796i)Λ(1−s)
Λ(s)=(=(812s/2ΓR(s+1)L(s)(−0.604+0.796i)Λ(1−s)
Degree: |
1 |
Conductor: |
812
= 22⋅7⋅29
|
Sign: |
−0.604+0.796i
|
Analytic conductor: |
87.2615 |
Root analytic conductor: |
87.2615 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ812(23,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 812, (1: ), −0.604+0.796i)
|
Particular Values
L(21) |
≈ |
0.1925805729+0.3879305579i |
L(21) |
≈ |
0.1925805729+0.3879305579i |
L(1) |
≈ |
0.6109691665+0.1048819133i |
L(1) |
≈ |
0.6109691665+0.1048819133i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 7 | 1 |
| 29 | 1 |
good | 3 | 1+(−0.826+0.563i)T |
| 5 | 1+(−0.733+0.680i)T |
| 11 | 1+(−0.365−0.930i)T |
| 13 | 1+(0.623−0.781i)T |
| 17 | 1+(−0.5+0.866i)T |
| 19 | 1+(−0.826−0.563i)T |
| 23 | 1+(−0.955+0.294i)T |
| 31 | 1+(−0.955−0.294i)T |
| 37 | 1+(0.365−0.930i)T |
| 41 | 1+T |
| 43 | 1+(0.222−0.974i)T |
| 47 | 1+(0.988+0.149i)T |
| 53 | 1+(0.955+0.294i)T |
| 59 | 1+(0.5−0.866i)T |
| 61 | 1+(0.0747+0.997i)T |
| 67 | 1+(0.988−0.149i)T |
| 71 | 1+(−0.623+0.781i)T |
| 73 | 1+(−0.733−0.680i)T |
| 79 | 1+(−0.365+0.930i)T |
| 83 | 1+(0.900−0.433i)T |
| 89 | 1+(−0.733+0.680i)T |
| 97 | 1+(−0.900+0.433i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−21.85538692604712994390047741373, −20.84825931330324661409212618109, −20.14442163150450915297688443184, −19.27027273132137600191400898822, −18.43623599901519309380708585325, −17.838230846968046158022119464104, −16.79469853133897629153209678786, −16.236861065422450506404511615845, −15.547634045946904545002381993655, −14.39909037351179816666810235044, −13.31172119122204261718521658561, −12.660828152704930902549913765386, −11.93284533459945888195268770720, −11.282154601932859989462339765767, −10.34982479521299849924878491191, −9.23748182407340856714040038362, −8.24351086478954876639586449399, −7.43920121934246021729997465581, −6.64118791593643889986839238147, −5.63553474972288508657785486244, −4.6153437791289826976048305172, −4.06473103675278483600872440532, −2.327096168792506863074587509531, −1.36848330273667108334205671175, −0.1675910863949312598136110546,
0.70528579364176598154453339707, 2.4596420434415499734376008310, 3.71666245009398318240934516438, 4.11888319850184823352800293540, 5.577947974223020463539007646110, 6.09343170217812947516546526294, 7.132240279948895308322731906617, 8.16058263148297404050847596494, 9.00379892287334772089140280785, 10.35209330635701477328150012402, 10.82738686747322918032069306512, 11.35799095691726555328944136180, 12.4106239011243767344029735193, 13.21616340611963759557700784389, 14.41101347306453665891426976707, 15.30133010417097848064149705281, 15.77982119041340276265694446883, 16.54151536497249734730214805397, 17.58094905381142459447997956907, 18.18365056550471384709351017881, 19.057685408105389653488635880062, 19.88202405973071447953208394251, 20.83156512284934947560061908708, 21.86935231003756948842863104936, 22.08519248203468849407412483907