L(s) = 1 | + (−0.915 + 0.401i)2-s + (0.546 + 0.837i)3-s + (0.677 − 0.735i)4-s + (0.986 + 0.164i)5-s + (−0.837 − 0.546i)6-s + (−0.614 + 0.789i)7-s + (−0.324 + 0.945i)8-s + (−0.401 + 0.915i)9-s + (−0.969 + 0.245i)10-s + (0.879 − 0.475i)11-s + (0.986 + 0.164i)12-s + (−0.164 + 0.986i)13-s + (0.245 − 0.969i)14-s + (0.401 + 0.915i)15-s + (−0.0825 − 0.996i)16-s + (−0.986 − 0.164i)17-s + ⋯ |
L(s) = 1 | + (−0.915 + 0.401i)2-s + (0.546 + 0.837i)3-s + (0.677 − 0.735i)4-s + (0.986 + 0.164i)5-s + (−0.837 − 0.546i)6-s + (−0.614 + 0.789i)7-s + (−0.324 + 0.945i)8-s + (−0.401 + 0.915i)9-s + (−0.969 + 0.245i)10-s + (0.879 − 0.475i)11-s + (0.986 + 0.164i)12-s + (−0.164 + 0.986i)13-s + (0.245 − 0.969i)14-s + (0.401 + 0.915i)15-s + (−0.0825 − 0.996i)16-s + (−0.986 − 0.164i)17-s + ⋯ |
Λ(s)=(=(229s/2ΓR(s+1)L(s)(−0.996−0.0861i)Λ(1−s)
Λ(s)=(=(229s/2ΓR(s+1)L(s)(−0.996−0.0861i)Λ(1−s)
Degree: |
1 |
Conductor: |
229
|
Sign: |
−0.996−0.0861i
|
Analytic conductor: |
24.6094 |
Root analytic conductor: |
24.6094 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ229(227,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 229, (1: ), −0.996−0.0861i)
|
Particular Values
L(21) |
≈ |
−0.04905557393+1.137062644i |
L(21) |
≈ |
−0.04905557393+1.137062644i |
L(1) |
≈ |
0.6598720915+0.5597409888i |
L(1) |
≈ |
0.6598720915+0.5597409888i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 229 | 1 |
good | 2 | 1+(−0.915+0.401i)T |
| 3 | 1+(0.546+0.837i)T |
| 5 | 1+(0.986+0.164i)T |
| 7 | 1+(−0.614+0.789i)T |
| 11 | 1+(0.879−0.475i)T |
| 13 | 1+(−0.164+0.986i)T |
| 17 | 1+(−0.986−0.164i)T |
| 19 | 1+(−0.986+0.164i)T |
| 23 | 1+(−0.969−0.245i)T |
| 29 | 1+(0.614−0.789i)T |
| 31 | 1+(0.475+0.879i)T |
| 37 | 1+(−0.0825+0.996i)T |
| 41 | 1+(0.915−0.401i)T |
| 43 | 1+(−0.0825+0.996i)T |
| 47 | 1+(−0.915−0.401i)T |
| 53 | 1+(0.546+0.837i)T |
| 59 | 1+(−0.996+0.0825i)T |
| 61 | 1+(−0.401+0.915i)T |
| 67 | 1+(−0.915−0.401i)T |
| 71 | 1+(0.879+0.475i)T |
| 73 | 1+(0.324−0.945i)T |
| 79 | 1+(0.614+0.789i)T |
| 83 | 1+(−0.0825−0.996i)T |
| 89 | 1−iT |
| 97 | 1+(−0.945+0.324i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−25.76817953148810219532236146812, −25.0056935604634254377777543874, −24.24610850544467898727671621075, −22.79687567949567968531679480211, −21.75885358642811421483410386338, −20.533234113447210446025988095104, −19.91260057125772754215782121136, −19.30488926475791652388451079407, −17.86243143096065125636772621534, −17.60342321796952638635946164581, −16.622187531925435451367768431101, −15.19768789967504023222731713199, −13.937653058473785976160872421179, −12.97201757029250998116598634389, −12.39337248536618232454166020304, −10.8769933948890276996981306090, −9.85443827875458376775541353867, −9.09336674879372244384713247711, −8.03048205322681360573129170717, −6.854250760886880195305968017828, −6.237361368819210981298567913777, −3.96644341734941614589063583808, −2.628029655918101129394473275775, −1.65638416919573853723892636171, −0.43463270778145510773530029874,
1.86859123062587291525117954261, 2.780593222184314247845277602944, 4.541133364787728090645486661851, 6.03213778537063548279313948413, 6.5978117860025751380952118049, 8.42879362150924884974644393329, 9.09689635892592858037189982913, 9.74742544675966573008601723179, 10.709773053844803856503135713097, 11.88614561858664320719418415057, 13.64510578436267579440918530986, 14.42952775094207087304503840430, 15.35757553301193084614184579066, 16.32465412676701714453622596138, 17.01402213122328747257069024919, 18.133182938204450489996472493128, 19.217938905175056789773143927984, 19.77085313570928063411240725384, 21.14772374868589393059156077132, 21.70835725314238068604272215830, 22.70687520345663557753748193730, 24.44297202510128417354892290909, 24.974989894432369509273548046077, 25.90691130276667859916585549273, 26.38029051522484487444260982750