L(s) = 1 | + (0.727 − 0.686i)2-s + (0.0581 − 0.998i)4-s + (−0.116 − 0.993i)5-s + (−0.448 + 0.893i)7-s + (−0.642 − 0.766i)8-s + (−0.766 − 0.642i)10-s + (0.918 − 0.396i)11-s + (0.286 + 0.957i)14-s + (−0.993 − 0.116i)16-s + (0.173 − 0.984i)17-s + (0.984 − 0.173i)19-s + (−0.998 + 0.0581i)20-s + (0.396 − 0.918i)22-s + (0.893 − 0.448i)23-s + (−0.973 + 0.230i)25-s + ⋯ |
L(s) = 1 | + (0.727 − 0.686i)2-s + (0.0581 − 0.998i)4-s + (−0.116 − 0.993i)5-s + (−0.448 + 0.893i)7-s + (−0.642 − 0.766i)8-s + (−0.766 − 0.642i)10-s + (0.918 − 0.396i)11-s + (0.286 + 0.957i)14-s + (−0.993 − 0.116i)16-s + (0.173 − 0.984i)17-s + (0.984 − 0.173i)19-s + (−0.998 + 0.0581i)20-s + (0.396 − 0.918i)22-s + (0.893 − 0.448i)23-s + (−0.973 + 0.230i)25-s + ⋯ |
Λ(s)=(=(1053s/2ΓR(s)L(s)(−0.900−0.434i)Λ(1−s)
Λ(s)=(=(1053s/2ΓR(s)L(s)(−0.900−0.434i)Λ(1−s)
Degree: |
1 |
Conductor: |
1053
= 34⋅13
|
Sign: |
−0.900−0.434i
|
Analytic conductor: |
4.89011 |
Root analytic conductor: |
4.89011 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1053(1019,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 1053, (0: ), −0.900−0.434i)
|
Particular Values
L(21) |
≈ |
0.4319799019−1.891133727i |
L(21) |
≈ |
0.4319799019−1.891133727i |
L(1) |
≈ |
1.094112335−0.9487987528i |
L(1) |
≈ |
1.094112335−0.9487987528i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 13 | 1 |
good | 2 | 1+(0.727−0.686i)T |
| 5 | 1+(−0.116−0.993i)T |
| 7 | 1+(−0.448+0.893i)T |
| 11 | 1+(0.918−0.396i)T |
| 17 | 1+(0.173−0.984i)T |
| 19 | 1+(0.984−0.173i)T |
| 23 | 1+(0.893−0.448i)T |
| 29 | 1+(0.286−0.957i)T |
| 31 | 1+(−0.549+0.835i)T |
| 37 | 1+(−0.342−0.939i)T |
| 41 | 1+(−0.727−0.686i)T |
| 43 | 1+(−0.597+0.802i)T |
| 47 | 1+(0.549+0.835i)T |
| 53 | 1+(0.5−0.866i)T |
| 59 | 1+(−0.918−0.396i)T |
| 61 | 1+(−0.0581−0.998i)T |
| 67 | 1+(−0.957+0.286i)T |
| 71 | 1+(−0.642+0.766i)T |
| 73 | 1+(0.642+0.766i)T |
| 79 | 1+(−0.686−0.727i)T |
| 83 | 1+(0.727−0.686i)T |
| 89 | 1+(−0.642−0.766i)T |
| 97 | 1+(−0.116+0.993i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−22.25001982027779823426176889761, −21.34482042155717153602343795476, −20.270324631019267882878733257004, −19.69412934229826411586492024901, −18.647628616835440395798241396135, −17.7876512355595047867701967677, −16.91693653999165845860932514149, −16.51820371553995315924960069509, −15.12735020686072278827112914191, −15.05232557849157032783916061863, −13.896399498862589293257187689812, −13.54879975576886650477307091745, −12.42755013177650054339111648675, −11.684411952038979474592335283909, −10.77750263449023730541905579716, −9.90980095979584962826847592018, −8.874684953971608246048686473803, −7.69254458523652869152757061701, −7.08259922815284361274521459507, −6.51863966554143765980133746878, −5.595474625623098514506595351849, −4.40033872501894186267812856477, −3.584525777578290147464218146507, −3.062487644784988914223839766, −1.555568923955259663835138154429,
0.64915963266124498905062060557, 1.67055621773747460005069536738, 2.83739632136850266847396492050, 3.61031134603136025528302958162, 4.7240558457949577284193438401, 5.35719809650905593744060880464, 6.1697370621010229246693551074, 7.18945629089532990892879245674, 8.65584107191622019067746049291, 9.22161841709110356086138118522, 9.84614889908052677292169268590, 11.16786480128226626289199367643, 11.86164078885456512877210535334, 12.346538471342916442293723917081, 13.18425882270644783288454794812, 13.93737661172613242780217755383, 14.75129912278339934286712940136, 15.82450543772515118239819438349, 16.14287729969802925296160011721, 17.28650942958784013116011419228, 18.3444029073607962695355769205, 19.10190098046030407164258524844, 19.758626054910318347379831206763, 20.4713316922022079986854227078, 21.21133066053041501904114094057