L(s) = 1 | + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.809 − 0.587i)7-s + (−0.809 − 0.587i)8-s + (−0.309 − 0.951i)13-s + (0.809 + 0.587i)14-s + (0.309 − 0.951i)16-s + (0.309 − 0.951i)17-s + (−0.809 − 0.587i)19-s + 23-s + (0.809 − 0.587i)26-s + (−0.309 + 0.951i)28-s + (0.809 − 0.587i)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 + (0.809 − 0.587i)7-s + (−0.809 − 0.587i)8-s + (−0.309 − 0.951i)13-s + (0.809 + 0.587i)14-s + (0.309 − 0.951i)16-s + (0.309 − 0.951i)17-s + (−0.809 − 0.587i)19-s + 23-s + (0.809 − 0.587i)26-s + (−0.309 + 0.951i)28-s + (0.809 − 0.587i)29-s + (0.309 + 0.951i)31-s + 32-s + ⋯ |
Λ(s)=(=(165s/2ΓR(s+1)L(s)(0.999+0.0237i)Λ(1−s)
Λ(s)=(=(165s/2ΓR(s+1)L(s)(0.999+0.0237i)Λ(1−s)
Degree: |
1 |
Conductor: |
165
= 3⋅5⋅11
|
Sign: |
0.999+0.0237i
|
Analytic conductor: |
17.7317 |
Root analytic conductor: |
17.7317 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ165(59,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 165, (1: ), 0.999+0.0237i)
|
Particular Values
L(21) |
≈ |
1.842202984+0.02189922591i |
L(21) |
≈ |
1.842202984+0.02189922591i |
L(1) |
≈ |
1.180010477+0.3156524338i |
L(1) |
≈ |
1.180010477+0.3156524338i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 5 | 1 |
| 11 | 1 |
good | 2 | 1+(0.309+0.951i)T |
| 7 | 1+(0.809−0.587i)T |
| 13 | 1+(−0.309−0.951i)T |
| 17 | 1+(0.309−0.951i)T |
| 19 | 1+(−0.809−0.587i)T |
| 23 | 1+T |
| 29 | 1+(0.809−0.587i)T |
| 31 | 1+(0.309+0.951i)T |
| 37 | 1+(0.809−0.587i)T |
| 41 | 1+(0.809+0.587i)T |
| 43 | 1−T |
| 47 | 1+(−0.809−0.587i)T |
| 53 | 1+(0.309+0.951i)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+(0.809−0.587i)T |
| 79 | 1+(0.309+0.951i)T |
| 83 | 1+(0.309−0.951i)T |
| 89 | 1−T |
| 97 | 1+(−0.309−0.951i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−27.614893119725069111727354825, −26.83243495680684989844254730658, −25.52097622552621008047627282981, −24.218956406478270990165688160401, −23.55433176746190092545529767945, −22.37070488875462086967793106542, −21.32239578632439383017542722511, −20.97861580673976930855888011369, −19.5211187522528427653214144076, −18.857226261705920929791695381642, −17.821626195002359017496144288768, −16.749660841192857323870648030997, −15.00020496747456639811077963069, −14.50221901273906989929685545704, −13.20346542761596825317024240470, −12.17927629551741866778475626675, −11.35273895596605748412501472526, −10.31280878021595972422671289341, −9.090879794952387512823221569, −8.14898126453503682677819704259, −6.30168308430869655207464589874, −5.04969840672151703782358970207, −4.0388042415341970205823868748, −2.49820483734751186274472058341, −1.39666831725325105993889984514,
0.675309615403209746515152422862, 2.94401963387254687331720241075, 4.468633205926962461787359406790, 5.27064677035145147833242082612, 6.71979750019681919352456648956, 7.6815253889625976770405007652, 8.61560091843868512512955243955, 9.97019420836844925324660640830, 11.29640661899691209183358456495, 12.62652424648961540220737741404, 13.60820590606683415381143902321, 14.57262681070385785048833071420, 15.39408347671030352527395232758, 16.56954721245639941494306116515, 17.481287659055105882675767517307, 18.1719210670661173625338457347, 19.6094994413403636470873426338, 20.86219841276942533675034378079, 21.69826685353897400514180693792, 23.00751039202771952144827635642, 23.450834401524765493227148684554, 24.74657843366136013348333109870, 25.18340827252035669490682651391, 26.58269143641430988556633112200, 27.136583698082379899401535044