L(s) = 1 | + (−0.597 − 0.802i)2-s + (−0.286 + 0.957i)4-s + (0.835 − 0.549i)5-s + (−0.686 + 0.727i)7-s + (0.939 − 0.342i)8-s + (−0.939 − 0.342i)10-s + (−0.893 + 0.448i)11-s + (0.396 + 0.918i)13-s + (0.993 + 0.116i)14-s + (−0.835 − 0.549i)16-s + (−0.766 + 0.642i)17-s + (0.766 + 0.642i)19-s + (0.286 + 0.957i)20-s + (0.893 + 0.448i)22-s + (0.686 + 0.727i)23-s + ⋯ |
L(s) = 1 | + (−0.597 − 0.802i)2-s + (−0.286 + 0.957i)4-s + (0.835 − 0.549i)5-s + (−0.686 + 0.727i)7-s + (0.939 − 0.342i)8-s + (−0.939 − 0.342i)10-s + (−0.893 + 0.448i)11-s + (0.396 + 0.918i)13-s + (0.993 + 0.116i)14-s + (−0.835 − 0.549i)16-s + (−0.766 + 0.642i)17-s + (0.766 + 0.642i)19-s + (0.286 + 0.957i)20-s + (0.893 + 0.448i)22-s + (0.686 + 0.727i)23-s + ⋯ |
Λ(s)=(=(81s/2ΓR(s+1)L(s)(0.824+0.565i)Λ(1−s)
Λ(s)=(=(81s/2ΓR(s+1)L(s)(0.824+0.565i)Λ(1−s)
Degree: |
1 |
Conductor: |
81
= 34
|
Sign: |
0.824+0.565i
|
Analytic conductor: |
8.70465 |
Root analytic conductor: |
8.70465 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ81(68,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 81, (1: ), 0.824+0.565i)
|
Particular Values
L(21) |
≈ |
0.9760299288+0.3025466108i |
L(21) |
≈ |
0.9760299288+0.3025466108i |
L(1) |
≈ |
0.8079441753−0.07197521243i |
L(1) |
≈ |
0.8079441753−0.07197521243i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
good | 2 | 1+(−0.597−0.802i)T |
| 5 | 1+(0.835−0.549i)T |
| 7 | 1+(−0.686+0.727i)T |
| 11 | 1+(−0.893+0.448i)T |
| 13 | 1+(0.396+0.918i)T |
| 17 | 1+(−0.766+0.642i)T |
| 19 | 1+(0.766+0.642i)T |
| 23 | 1+(0.686+0.727i)T |
| 29 | 1+(0.993−0.116i)T |
| 31 | 1+(0.973−0.230i)T |
| 37 | 1+(0.173+0.984i)T |
| 41 | 1+(−0.597+0.802i)T |
| 43 | 1+(−0.0581+0.998i)T |
| 47 | 1+(−0.973−0.230i)T |
| 53 | 1+(0.5+0.866i)T |
| 59 | 1+(−0.893−0.448i)T |
| 61 | 1+(−0.286−0.957i)T |
| 67 | 1+(−0.993−0.116i)T |
| 71 | 1+(0.939+0.342i)T |
| 73 | 1+(−0.939+0.342i)T |
| 79 | 1+(0.597+0.802i)T |
| 83 | 1+(−0.597−0.802i)T |
| 89 | 1+(0.939−0.342i)T |
| 97 | 1+(−0.835−0.549i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−30.55463942478311568961383795761, −29.19361747689156514954786803495, −28.6959252748132902641341021272, −26.99384793893313949832167616244, −26.36646503719704713137657689030, −25.4554003166452978293543696038, −24.460467170939492778447944503365, −23.12069585337849489293225532565, −22.37906745305228947569484813614, −20.73279498072467112898452206828, −19.506557378882574131707678850, −18.27108135323765144949354798261, −17.58405527164533911352253852073, −16.28922451668678074340387688471, −15.37388123011043664227372819935, −13.862211143333380928365328818428, −13.27021246328645089972948921031, −10.801563373123930192418421353665, −10.14416526923966496518898754766, −8.84865621804764099853288079429, −7.34002926327895227342633471542, −6.36143218570510640053509928465, −5.10118949560273703843638177535, −2.84394147514700377485976065657, −0.61684017480868729454737131053,
1.57357992113764614169910904856, 2.89286037731367279932198220096, 4.75632755644399388928709779207, 6.408036659739737929809163459415, 8.24718156052587009206979184742, 9.34232808102542672851006307127, 10.14987481997479060559164892399, 11.69810466184592313815299820453, 12.82857755815031678507427159084, 13.620657539978405722283504461513, 15.65641513437985504915760064119, 16.75132118564639655137684867754, 17.90306449617602486728740697731, 18.78002238721492769097944655769, 19.98237682009209999941887068652, 21.14821910864311992802782688073, 21.741903929909234895339865303299, 23.08708291159586995461294210357, 24.75279172770236171880721113663, 25.7226496993346751193287965480, 26.51199410488068123239624102965, 28.08692634695840246613963768419, 28.77688474477087338887608115724, 29.283562240495709088682048521451, 30.89310852338307113948718023381