L(s) = 1 | + (−0.5 + 0.866i)7-s + (0.978 + 0.207i)11-s + (−0.978 + 0.207i)13-s + (−0.809 − 0.587i)17-s + (−0.809 − 0.587i)19-s + (−0.669 + 0.743i)23-s + (−0.104 − 0.994i)29-s + (0.104 − 0.994i)31-s + (0.309 − 0.951i)37-s + (0.978 − 0.207i)41-s + (0.5 − 0.866i)43-s + (0.104 + 0.994i)47-s + (−0.5 − 0.866i)49-s + (0.809 − 0.587i)53-s + (0.978 − 0.207i)59-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)7-s + (0.978 + 0.207i)11-s + (−0.978 + 0.207i)13-s + (−0.809 − 0.587i)17-s + (−0.809 − 0.587i)19-s + (−0.669 + 0.743i)23-s + (−0.104 − 0.994i)29-s + (0.104 − 0.994i)31-s + (0.309 − 0.951i)37-s + (0.978 − 0.207i)41-s + (0.5 − 0.866i)43-s + (0.104 + 0.994i)47-s + (−0.5 − 0.866i)49-s + (0.809 − 0.587i)53-s + (0.978 − 0.207i)59-s + ⋯ |
Λ(s)=(=(1800s/2ΓR(s)L(s)(0.704−0.709i)Λ(1−s)
Λ(s)=(=(1800s/2ΓR(s)L(s)(0.704−0.709i)Λ(1−s)
Degree: |
1 |
Conductor: |
1800
= 23⋅32⋅52
|
Sign: |
0.704−0.709i
|
Analytic conductor: |
8.35916 |
Root analytic conductor: |
8.35916 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1800(59,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 1800, (0: ), 0.704−0.709i)
|
Particular Values
L(21) |
≈ |
1.027103232−0.4275418108i |
L(21) |
≈ |
1.027103232−0.4275418108i |
L(1) |
≈ |
0.9291205232+0.01275877398i |
L(1) |
≈ |
0.9291205232+0.01275877398i |
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+(0.978+0.207i)T |
| 13 | 1+(−0.978+0.207i)T |
| 17 | 1+(−0.809−0.587i)T |
| 19 | 1+(−0.809−0.587i)T |
| 23 | 1+(−0.669+0.743i)T |
| 29 | 1+(−0.104−0.994i)T |
| 31 | 1+(0.104−0.994i)T |
| 37 | 1+(0.309−0.951i)T |
| 41 | 1+(0.978−0.207i)T |
| 43 | 1+(0.5−0.866i)T |
| 47 | 1+(0.104+0.994i)T |
| 53 | 1+(0.809−0.587i)T |
| 59 | 1+(0.978−0.207i)T |
| 61 | 1+(0.978+0.207i)T |
| 67 | 1+(0.104−0.994i)T |
| 71 | 1+(−0.809+0.587i)T |
| 73 | 1+(−0.309−0.951i)T |
| 79 | 1+(0.104+0.994i)T |
| 83 | 1+(0.913−0.406i)T |
| 89 | 1+(−0.309−0.951i)T |
| 97 | 1+(0.104+0.994i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−20.03853671122189671145771296540, −19.670688440562386609363768207911, −19.04052232071214029554931070513, −17.92731292120082453795463679784, −17.298740090560240027174196145279, −16.60734543828966327967215219700, −16.100476344505751558332479800060, −14.8516932807257989536336305264, −14.510324392796024776313579682884, −13.61816559209301317164090071783, −12.77119056947466925285071462133, −12.23716396424071641136770471815, −11.22875001394572572167303917589, −10.41804093712964388965940813103, −9.89559747430292640417078500563, −8.894763985969988261355380670560, −8.211262411375715411819305989737, −7.11680407711391294733819429750, −6.62039012662900805406745888176, −5.79231724724445876002881566850, −4.523138183247230619116042519941, −4.0443618780320481671612202078, −3.061793687184466648181343508199, −2.006246052062899966683307822507, −0.923563628606952795530566917817,
0.46775391537895691160162973597, 2.16024702919748377134767484424, 2.430905069690093568328756010978, 3.814668388051827791625945165543, 4.47128051921847023077438621212, 5.53346582810865885078857534307, 6.2893319164555092307168450603, 7.030207963806101496864589238223, 7.88790096353740207573072698115, 9.1233719992353042320595914145, 9.260418739375843233769694739555, 10.18054115004643135334162245186, 11.384140743432060523023475862949, 11.797456664299385171419709085735, 12.63413157501773734330200706706, 13.33310153695511774134292067936, 14.28133591335174186887982292930, 14.972618415805690838209311120669, 15.634983138478694403978731367608, 16.35917890821525880774177028709, 17.36394714080026982658609878088, 17.6889158702951476348453124455, 18.80466309134208535833637181677, 19.44174573362615587830329305945, 19.83835293975147858064314457460