L(s) = 1 | + (0.891 − 0.453i)2-s + (0.760 + 0.649i)3-s + (0.587 − 0.809i)4-s + (0.522 + 0.852i)5-s + (0.972 + 0.233i)6-s + (0.760 − 0.649i)7-s + (0.156 − 0.987i)8-s + (0.156 + 0.987i)9-s + (0.852 + 0.522i)10-s + (0.923 − 0.382i)11-s + (0.972 − 0.233i)12-s + i·13-s + (0.382 − 0.923i)14-s + (−0.156 + 0.987i)15-s + (−0.309 − 0.951i)16-s + ⋯ |
L(s) = 1 | + (0.891 − 0.453i)2-s + (0.760 + 0.649i)3-s + (0.587 − 0.809i)4-s + (0.522 + 0.852i)5-s + (0.972 + 0.233i)6-s + (0.760 − 0.649i)7-s + (0.156 − 0.987i)8-s + (0.156 + 0.987i)9-s + (0.852 + 0.522i)10-s + (0.923 − 0.382i)11-s + (0.972 − 0.233i)12-s + i·13-s + (0.382 − 0.923i)14-s + (−0.156 + 0.987i)15-s + (−0.309 − 0.951i)16-s + ⋯ |
Λ(s)=(=(1037s/2ΓR(s)L(s)(0.998−0.0457i)Λ(1−s)
Λ(s)=(=(1037s/2ΓR(s)L(s)(0.998−0.0457i)Λ(1−s)
Degree: |
1 |
Conductor: |
1037
= 17⋅61
|
Sign: |
0.998−0.0457i
|
Analytic conductor: |
4.81580 |
Root analytic conductor: |
4.81580 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1037(114,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 1037, (0: ), 0.998−0.0457i)
|
Particular Values
L(21) |
≈ |
4.196284694−0.09610282262i |
L(21) |
≈ |
4.196284694−0.09610282262i |
L(1) |
≈ |
2.614444788−0.09811349123i |
L(1) |
≈ |
2.614444788−0.09811349123i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 17 | 1 |
| 61 | 1 |
good | 2 | 1+(0.891−0.453i)T |
| 3 | 1+(0.760+0.649i)T |
| 5 | 1+(0.522+0.852i)T |
| 7 | 1+(0.760−0.649i)T |
| 11 | 1+(0.923−0.382i)T |
| 13 | 1+iT |
| 19 | 1+(−0.453−0.891i)T |
| 23 | 1+(−0.972−0.233i)T |
| 29 | 1+(0.382−0.923i)T |
| 31 | 1+(0.0784−0.996i)T |
| 37 | 1+(−0.0784+0.996i)T |
| 41 | 1+(−0.649−0.760i)T |
| 43 | 1+(−0.156+0.987i)T |
| 47 | 1+iT |
| 53 | 1+(−0.987−0.156i)T |
| 59 | 1+(0.453+0.891i)T |
| 67 | 1+(−0.587+0.809i)T |
| 71 | 1+(−0.972+0.233i)T |
| 73 | 1+(−0.972+0.233i)T |
| 79 | 1+(0.522−0.852i)T |
| 83 | 1+(−0.453−0.891i)T |
| 89 | 1+(0.309−0.951i)T |
| 97 | 1+(−0.0784+0.996i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−21.55812193894572547381323509325, −20.77870081990001507544156640514, −20.216779253019375328466842970604, −19.57008500676707413839470608964, −18.13437738454971991448367024388, −17.71529417522629769194877207234, −16.87988073921766266844449208875, −15.865839720132350349670498230043, −15.042554056255056836842810989273, −14.34695815699386896186245953979, −13.85403943899005727706241675877, −12.786517114147177065889416139869, −12.35540298222122919172144239765, −11.773267895449093130619369199105, −10.334588457646945240759911613149, −9.11874198198022257929458047525, −8.41382199582539139204312545154, −7.889219352050032491673665186262, −6.7808089649589516459498619652, −5.91578209424606433387656632031, −5.18035963032012680053098121486, −4.172151529217818735659148515010, −3.22460303060777134394086028604, −2.022589675295158801536942862329, −1.51449622586999298850002790659,
1.5198561605003809535524286133, 2.27432081456357607961182814753, 3.200970640685717127346799856623, 4.20935736463417224030721755430, 4.55021400884888362147547490777, 5.93717103918316213985817073372, 6.70465249692645724730736339246, 7.63226759960035726243829681830, 8.85263625329188659970479870217, 9.77817154233911197339846725904, 10.38284304187205817126533862524, 11.300113628932974893480512547057, 11.69106108442101229767681803884, 13.35273599010761756010825096582, 13.72828273657754478808592801842, 14.47508070347107440731273467306, 14.80034397662733469979833285532, 15.79451069125959112798815721407, 16.74391741306173407258401106303, 17.61072625373415480522156361932, 18.94062327418915203419289477004, 19.27238164111409979527049339739, 20.26884419297289115930008826067, 20.86607910644883460638926142672, 21.66269227700596850314267887382