L(s) = 1 | + (−0.793 − 0.607i)3-s + (−0.187 − 0.982i)5-s + (−0.762 − 0.647i)7-s + (0.260 + 0.965i)9-s + (−0.402 − 0.915i)11-s + (0.356 − 0.934i)13-s + (−0.448 + 0.893i)15-s + (0.617 − 0.786i)17-s + (−0.693 + 0.720i)19-s + (0.212 + 0.977i)21-s + (0.556 + 0.830i)23-s + (−0.929 + 0.368i)25-s + (0.379 − 0.925i)27-s + (−0.597 + 0.801i)29-s + (0.863 + 0.503i)31-s + ⋯ |
L(s) = 1 | + (−0.793 − 0.607i)3-s + (−0.187 − 0.982i)5-s + (−0.762 − 0.647i)7-s + (0.260 + 0.965i)9-s + (−0.402 − 0.915i)11-s + (0.356 − 0.934i)13-s + (−0.448 + 0.893i)15-s + (0.617 − 0.786i)17-s + (−0.693 + 0.720i)19-s + (0.212 + 0.977i)21-s + (0.556 + 0.830i)23-s + (−0.929 + 0.368i)25-s + (0.379 − 0.925i)27-s + (−0.597 + 0.801i)29-s + (0.863 + 0.503i)31-s + ⋯ |
Λ(s)=(=(1004s/2ΓR(s+1)L(s)(0.999+0.00750i)Λ(1−s)
Λ(s)=(=(1004s/2ΓR(s+1)L(s)(0.999+0.00750i)Λ(1−s)
Degree: |
1 |
Conductor: |
1004
= 22⋅251
|
Sign: |
0.999+0.00750i
|
Analytic conductor: |
107.894 |
Root analytic conductor: |
107.894 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1004(263,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 1004, (1: ), 0.999+0.00750i)
|
Particular Values
L(21) |
≈ |
0.5674710483+0.002129253800i |
L(21) |
≈ |
0.5674710483+0.002129253800i |
L(1) |
≈ |
0.5893773544−0.2923666089i |
L(1) |
≈ |
0.5893773544−0.2923666089i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 251 | 1 |
good | 3 | 1+(−0.793−0.607i)T |
| 5 | 1+(−0.187−0.982i)T |
| 7 | 1+(−0.762−0.647i)T |
| 11 | 1+(−0.402−0.915i)T |
| 13 | 1+(0.356−0.934i)T |
| 17 | 1+(0.617−0.786i)T |
| 19 | 1+(−0.693+0.720i)T |
| 23 | 1+(0.556+0.830i)T |
| 29 | 1+(−0.597+0.801i)T |
| 31 | 1+(0.863+0.503i)T |
| 37 | 1+(−0.837+0.546i)T |
| 41 | 1+(0.162−0.986i)T |
| 43 | 1+(−0.492−0.870i)T |
| 47 | 1+(0.637+0.770i)T |
| 53 | 1+(0.112+0.993i)T |
| 59 | 1+(−0.617−0.786i)T |
| 61 | 1+(−0.910+0.414i)T |
| 67 | 1+(−0.448−0.893i)T |
| 71 | 1+(0.778+0.627i)T |
| 73 | 1+(0.823+0.567i)T |
| 79 | 1+(−0.577+0.816i)T |
| 83 | 1+(0.0878+0.996i)T |
| 89 | 1+(−0.0376+0.999i)T |
| 97 | 1+(0.656+0.754i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−21.41597655713701920156369363273, −21.10724133798138456769688252915, −19.766238016045127918073008834105, −18.910784789249544711073711415614, −18.425575492863427463390912798178, −17.492399819127207478712710326304, −16.7266178059022395020752582365, −15.882532507549557898358571389660, −15.08472180085750457906682834612, −14.81350747119655202198648483857, −13.40678101720987726957090322726, −12.51255548376630215954091174606, −11.78986978548949231967817741146, −10.99178143841310028145724646482, −10.208093871348478993309870634793, −9.60657481770998261130506363916, −8.63051070190245764047445466030, −7.31634072720820560441298233847, −6.45844423311714942998790780796, −6.03798272500010556787557123597, −4.78188830977507716165627936498, −3.976363389163286021530213142686, −2.98988978035641449164179341110, −1.97344687811175219777094255521, −0.204521429549507119670133791990,
0.692918187094835288377184441, 1.36705030714407829698467253926, 2.97741195928927712022174228295, 3.90997670331458355543238799254, 5.20698142275620679982515056148, 5.62090781925954518500814599706, 6.656067134774701304715661680117, 7.612048318239277607762700388570, 8.26633778518306937975300068291, 9.32811358691679381625321885602, 10.42989823845697851217786826279, 10.95251789875590355585106125995, 12.15687066440585168474764596490, 12.56598084601082022699089971740, 13.48157329905345838631919929162, 13.84558860690985492733704914376, 15.57430019146996939443475887894, 16.034391045684914818226264364422, 16.95177067960633885965644198084, 17.18497090560552338817400489380, 18.45536771997072606862531465373, 19.01637915463881566432483156460, 19.825202411200172415776607919464, 20.65609720329501456106005287051, 21.39920838123296651503182702788