L(s) = 1 | + (−0.707 + 0.707i)2-s + (0.923 − 0.382i)3-s − i·4-s + (−0.382 + 0.923i)6-s + (−0.382 + 0.923i)7-s + (0.707 + 0.707i)8-s + (0.707 − 0.707i)9-s + (0.382 − 0.923i)11-s + (−0.382 − 0.923i)12-s + 13-s + (−0.382 − 0.923i)14-s − 16-s + i·18-s + (0.707 + 0.707i)19-s + i·21-s + (0.382 + 0.923i)22-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)2-s + (0.923 − 0.382i)3-s − i·4-s + (−0.382 + 0.923i)6-s + (−0.382 + 0.923i)7-s + (0.707 + 0.707i)8-s + (0.707 − 0.707i)9-s + (0.382 − 0.923i)11-s + (−0.382 − 0.923i)12-s + 13-s + (−0.382 − 0.923i)14-s − 16-s + i·18-s + (0.707 + 0.707i)19-s + i·21-s + (0.382 + 0.923i)22-s + ⋯ |
Λ(s)=(=(85s/2ΓR(s)L(s)(0.922+0.386i)Λ(1−s)
Λ(s)=(=(85s/2ΓR(s)L(s)(0.922+0.386i)Λ(1−s)
Degree: |
1 |
Conductor: |
85
= 5⋅17
|
Sign: |
0.922+0.386i
|
Analytic conductor: |
0.394738 |
Root analytic conductor: |
0.394738 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ85(58,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 85, (0: ), 0.922+0.386i)
|
Particular Values
L(21) |
≈ |
0.9153440286+0.1838165876i |
L(21) |
≈ |
0.9153440286+0.1838165876i |
L(1) |
≈ |
0.9629729231+0.1667101039i |
L(1) |
≈ |
0.9629729231+0.1667101039i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1 |
| 17 | 1 |
good | 2 | 1+(−0.707+0.707i)T |
| 3 | 1+(0.923−0.382i)T |
| 7 | 1+(−0.382+0.923i)T |
| 11 | 1+(0.382−0.923i)T |
| 13 | 1+T |
| 19 | 1+(0.707+0.707i)T |
| 23 | 1+(−0.923−0.382i)T |
| 29 | 1+(−0.923+0.382i)T |
| 31 | 1+(0.382+0.923i)T |
| 37 | 1+(−0.923+0.382i)T |
| 41 | 1+(−0.923−0.382i)T |
| 43 | 1+(−0.707−0.707i)T |
| 47 | 1−T |
| 53 | 1+(0.707−0.707i)T |
| 59 | 1+(−0.707+0.707i)T |
| 61 | 1+(0.923+0.382i)T |
| 67 | 1+iT |
| 71 | 1+(−0.382−0.923i)T |
| 73 | 1+(−0.382−0.923i)T |
| 79 | 1+(−0.382+0.923i)T |
| 83 | 1+(−0.707+0.707i)T |
| 89 | 1−iT |
| 97 | 1+(−0.382−0.923i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−30.46086225467897291674456773004, −29.87211484345893164568997686294, −28.33808722609564366757171051358, −27.589299056977549218198934356702, −26.17013534316152355455189828806, −26.072198900305498811638704289346, −24.69443015540158179839122647923, −22.94879109904264248986169386884, −21.8168896187675971020326870154, −20.48507812825529800837355366159, −20.14809787698167031356364142734, −19.03347652624144040646956144635, −17.802871817731663455393288912631, −16.55188619366019991226596211392, −15.48508552928735558890126003430, −13.83834688551077615783614421403, −13.0225402860067333200088001937, −11.405703658796288540761325182, −10.1143483925876720813396415097, −9.40633484419589703967285901171, −8.072908086531523831979301224615, −7.0084981767532850879113275372, −4.291399181366467096049173106058, −3.33643157289803732815401311268, −1.691482926362417444885704236761,
1.63821709722534523897851063561, 3.42053129634810597613967514474, 5.69419081218228498788741865719, 6.76917640314439923140342171327, 8.30570068255287426157807671635, 8.85468922881508184730955885268, 10.11731908490325898745576309317, 11.834609752335949660615392483294, 13.459502349675083138737016394737, 14.40105141544392212802098517723, 15.56861997693161397342861815166, 16.40398747024723385515677199199, 18.15649067663381311789609013316, 18.72073543975814796777023019145, 19.688716625135326199027415533566, 20.88136723636168541699612804860, 22.39164009731681268498756881142, 23.87218516935757756989925194731, 24.71666896192574438406384021808, 25.51365523857429334595703506864, 26.36871693050308580090879617432, 27.43145103862959100583319081446, 28.57629545318949943937993959361, 29.659874032527615353840679648497, 31.02270612493989696073537259282