L(s) = 1 | + (−0.848 + 0.529i)2-s + (0.438 − 0.898i)4-s + (−0.173 + 0.984i)7-s + (0.104 + 0.994i)8-s + (0.0348 + 0.999i)11-s + (−0.848 − 0.529i)13-s + (−0.374 − 0.927i)14-s + (−0.615 − 0.788i)16-s + (−0.913 − 0.406i)17-s + (−0.104 − 0.994i)19-s + (−0.559 − 0.829i)22-s + (−0.990 − 0.139i)23-s + 26-s + (0.809 + 0.587i)28-s + (−0.241 − 0.970i)29-s + ⋯ |
L(s) = 1 | + (−0.848 + 0.529i)2-s + (0.438 − 0.898i)4-s + (−0.173 + 0.984i)7-s + (0.104 + 0.994i)8-s + (0.0348 + 0.999i)11-s + (−0.848 − 0.529i)13-s + (−0.374 − 0.927i)14-s + (−0.615 − 0.788i)16-s + (−0.913 − 0.406i)17-s + (−0.104 − 0.994i)19-s + (−0.559 − 0.829i)22-s + (−0.990 − 0.139i)23-s + 26-s + (0.809 + 0.587i)28-s + (−0.241 − 0.970i)29-s + ⋯ |
Λ(s)=(=(675s/2ΓR(s)L(s)(0.00930−0.999i)Λ(1−s)
Λ(s)=(=(675s/2ΓR(s)L(s)(0.00930−0.999i)Λ(1−s)
Degree: |
1 |
Conductor: |
675
= 33⋅52
|
Sign: |
0.00930−0.999i
|
Analytic conductor: |
3.13468 |
Root analytic conductor: |
3.13468 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ675(139,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 675, (0: ), 0.00930−0.999i)
|
Particular Values
L(21) |
≈ |
0.2012827203−0.1994177621i |
L(21) |
≈ |
0.2012827203−0.1994177621i |
L(1) |
≈ |
0.5386109102+0.1028584145i |
L(1) |
≈ |
0.5386109102+0.1028584145i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 5 | 1 |
good | 2 | 1+(−0.848+0.529i)T |
| 7 | 1+(−0.173+0.984i)T |
| 11 | 1+(0.0348+0.999i)T |
| 13 | 1+(−0.848−0.529i)T |
| 17 | 1+(−0.913−0.406i)T |
| 19 | 1+(−0.104−0.994i)T |
| 23 | 1+(−0.990−0.139i)T |
| 29 | 1+(−0.241−0.970i)T |
| 31 | 1+(−0.719−0.694i)T |
| 37 | 1+(−0.669+0.743i)T |
| 41 | 1+(0.848+0.529i)T |
| 43 | 1+(0.939−0.342i)T |
| 47 | 1+(0.719−0.694i)T |
| 53 | 1+(0.809+0.587i)T |
| 59 | 1+(0.0348−0.999i)T |
| 61 | 1+(−0.882−0.469i)T |
| 67 | 1+(0.241−0.970i)T |
| 71 | 1+(−0.104+0.994i)T |
| 73 | 1+(−0.669−0.743i)T |
| 79 | 1+(−0.241−0.970i)T |
| 83 | 1+(0.997+0.0697i)T |
| 89 | 1+(0.669+0.743i)T |
| 97 | 1+(−0.961−0.275i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−22.759652246193054108200355833045, −21.92307193171473408606510958885, −21.249499971563297800247346311365, −20.30019249451425104109596555712, −19.60357422343143794412998072117, −19.080259820109606613932026912096, −18.00211668828549531815745463907, −17.31336186843605256902084821802, −16.4121915609024750023981196715, −16.05819347956319751553487876925, −14.54457061498612405079931082650, −13.76214058354000158154230726116, −12.755775034915239387687767522371, −11.98917713189856537895219054118, −10.869323686444185828085166381472, −10.50027406013580553985950696104, −9.413442869167588406822034333128, −8.64994307223101551731605732889, −7.627812731896061139335130973642, −6.94325493376608177605107255637, −5.84991902194875672384012735650, −4.230745983328097486261142542229, −3.59400125048563062680834886119, −2.34465881061300123609852210159, −1.257402745718686308965750592187,
0.18395037104452178084614403491, 2.0430389151494575289811355424, 2.57417524859053701606711584368, 4.463317050395295388143542793426, 5.35843189514603451490721400338, 6.29806904765350963081728291687, 7.21218676569625999724739351139, 8.003195431952804214979158988335, 9.11603886380615157296356835768, 9.56606568185475100308551296133, 10.54348111010646729154916444486, 11.595646094878680463674909418666, 12.35890035976335273263862215811, 13.46856522377336172261638968942, 14.655028782014617583876988652997, 15.35311891164860152765795017675, 15.75956104241353235077744911921, 16.97862686394468658813192920164, 17.69218564173921487896751706152, 18.258614018366458972414229644678, 19.20336128543622657009970160275, 19.96511035921903674962184756667, 20.59176732922214304077380915000, 21.95254825489419645859743753974, 22.49377096746491883919048147190