L(s) = 1 | + 7-s + 11-s − 13-s + 17-s − 19-s − 23-s − 29-s − 31-s − 37-s − 41-s + 43-s − 47-s + 49-s + 53-s + 59-s + 61-s + 67-s + 71-s − 73-s + 77-s − 79-s − 83-s − 89-s − 91-s − 97-s + ⋯ |
L(s) = 1 | + 7-s + 11-s − 13-s + 17-s − 19-s − 23-s − 29-s − 31-s − 37-s − 41-s + 43-s − 47-s + 49-s + 53-s + 59-s + 61-s + 67-s + 71-s − 73-s + 77-s − 79-s − 83-s − 89-s − 91-s − 97-s + ⋯ |
Λ(s)=(=(60s/2ΓR(s)L(s)Λ(1−s)
Λ(s)=(=(60s/2ΓR(s)L(s)Λ(1−s)
Degree: |
1 |
Conductor: |
60
= 22⋅3⋅5
|
Sign: |
1
|
Analytic conductor: |
0.278638 |
Root analytic conductor: |
0.278638 |
Motivic weight: |
0 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
χ60(59,⋅)
|
Primitive: |
yes
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(1, 60, (0: ), 1)
|
Particular Values
L(21) |
≈ |
0.9618797159 |
L(21) |
≈ |
0.9618797159 |
L(1) |
≈ |
1.065554320 |
L(1) |
≈ |
1.065554320 |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
| 5 | 1 |
good | 7 | 1+T |
| 11 | 1+T |
| 13 | 1−T |
| 17 | 1+T |
| 19 | 1−T |
| 23 | 1−T |
| 29 | 1−T |
| 31 | 1−T |
| 37 | 1−T |
| 41 | 1−T |
| 43 | 1+T |
| 47 | 1−T |
| 53 | 1+T |
| 59 | 1+T |
| 61 | 1+T |
| 67 | 1+T |
| 71 | 1+T |
| 73 | 1−T |
| 79 | 1−T |
| 83 | 1−T |
| 89 | 1−T |
| 97 | 1−T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−32.58262270514236334702277741913, −31.56412029546067390492273146917, −30.28370271838972914813392774493, −29.56153792303208874429096790028, −27.86125861653568212621604974582, −27.34677171069150831416030737867, −25.89354093744892243070839471337, −24.673927390561852209576229240424, −23.800425575409428011559961836279, −22.36432718487739382101850758381, −21.328431099039773488357789758822, −20.11799276794188069833039241236, −18.9285284080218204999053966311, −17.554960636106842748012592192039, −16.66706135687040887534297750505, −14.9070292412479863596844040316, −14.21207272889969396426157491711, −12.45813508004805044439217178808, −11.39941638239152605493477803422, −9.94252189246792938879975776348, −8.485164567234391764673881624397, −7.166280912138318471719556203672, −5.46144112265697581553500848609, −3.985806388187321605499007281285, −1.88060641687791880942488827805,
1.88060641687791880942488827805, 3.985806388187321605499007281285, 5.46144112265697581553500848609, 7.166280912138318471719556203672, 8.485164567234391764673881624397, 9.94252189246792938879975776348, 11.39941638239152605493477803422, 12.45813508004805044439217178808, 14.21207272889969396426157491711, 14.9070292412479863596844040316, 16.66706135687040887534297750505, 17.554960636106842748012592192039, 18.9285284080218204999053966311, 20.11799276794188069833039241236, 21.328431099039773488357789758822, 22.36432718487739382101850758381, 23.800425575409428011559961836279, 24.673927390561852209576229240424, 25.89354093744892243070839471337, 27.34677171069150831416030737867, 27.86125861653568212621604974582, 29.56153792303208874429096790028, 30.28370271838972914813392774493, 31.56412029546067390492273146917, 32.58262270514236334702277741913