L(s) = 1 | + (0.933 − 0.358i)2-s + (0.481 + 0.876i)3-s + (0.742 − 0.669i)4-s + (0.129 − 0.991i)5-s + (0.763 + 0.645i)6-s + (0.490 − 0.871i)7-s + (0.453 − 0.891i)8-s + (−0.536 + 0.843i)9-s + (−0.234 − 0.972i)10-s + (0.835 − 0.549i)11-s + (0.944 + 0.328i)12-s + (−0.140 − 0.990i)13-s + (0.145 − 0.989i)14-s + (0.931 − 0.363i)15-s + (0.103 − 0.994i)16-s + (0.999 + 0.0186i)17-s + ⋯ |
L(s) = 1 | + (0.933 − 0.358i)2-s + (0.481 + 0.876i)3-s + (0.742 − 0.669i)4-s + (0.129 − 0.991i)5-s + (0.763 + 0.645i)6-s + (0.490 − 0.871i)7-s + (0.453 − 0.891i)8-s + (−0.536 + 0.843i)9-s + (−0.234 − 0.972i)10-s + (0.835 − 0.549i)11-s + (0.944 + 0.328i)12-s + (−0.140 − 0.990i)13-s + (0.145 − 0.989i)14-s + (0.931 − 0.363i)15-s + (0.103 − 0.994i)16-s + (0.999 + 0.0186i)17-s + ⋯ |
Λ(s)=(=(4729s/2ΓR(s)L(s)(−0.148−0.988i)Λ(1−s)
Λ(s)=(=(4729s/2ΓR(s)L(s)(−0.148−0.988i)Λ(1−s)
Degree: |
1 |
Conductor: |
4729
|
Sign: |
−0.148−0.988i
|
Analytic conductor: |
21.9613 |
Root analytic conductor: |
21.9613 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ4729(1022,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 4729, (0: ), −0.148−0.988i)
|
Particular Values
L(21) |
≈ |
3.080993321−3.579884233i |
L(21) |
≈ |
3.080993321−3.579884233i |
L(1) |
≈ |
2.245608188−0.9691640194i |
L(1) |
≈ |
2.245608188−0.9691640194i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 4729 | 1 |
good | 2 | 1+(0.933−0.358i)T |
| 3 | 1+(0.481+0.876i)T |
| 5 | 1+(0.129−0.991i)T |
| 7 | 1+(0.490−0.871i)T |
| 11 | 1+(0.835−0.549i)T |
| 13 | 1+(−0.140−0.990i)T |
| 17 | 1+(0.999+0.0186i)T |
| 19 | 1+(−0.0132−0.999i)T |
| 23 | 1+(0.402+0.915i)T |
| 29 | 1+(0.358−0.933i)T |
| 31 | 1+(−0.905+0.424i)T |
| 37 | 1+(0.0610+0.998i)T |
| 41 | 1+(0.417−0.908i)T |
| 43 | 1+(−0.730+0.683i)T |
| 47 | 1+(−0.744+0.667i)T |
| 53 | 1+(0.100−0.994i)T |
| 59 | 1+(0.326−0.945i)T |
| 61 | 1+(0.981−0.192i)T |
| 67 | 1+(−0.606+0.795i)T |
| 71 | 1+(−0.643+0.765i)T |
| 73 | 1+(−0.114+0.993i)T |
| 79 | 1+(0.866+0.5i)T |
| 83 | 1+(0.988+0.148i)T |
| 89 | 1+(−0.231+0.972i)T |
| 97 | 1+(−0.490+0.871i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−18.19178511658403693540083246466, −17.95937854489077523767049431646, −16.79688466082791170669831115671, −16.45073557749437417709606455855, −15.133614268594599972818895751898, −14.771975779409136333127637546825, −14.38778946537984388854438107329, −13.959387796440870664813241477649, −12.95919168916016707730490164690, −12.227201172975024730378962417422, −11.9211718515218400443233978682, −11.26345753955590274506228277413, −10.316221443935870689677594787279, −9.255621992580864817876827254605, −8.64505484745770015527986910067, −7.69638463364773615078972726734, −7.28257273331969103503742899957, −6.501036871251169220621965917450, −6.07495757024391071871330311707, −5.26500988608440678239819049356, −4.226639498022629894216596174247, −3.464644806985911032812802751387, −2.80363823259807162459576007135, −1.91832783848993049237908460752, −1.63217811976102892894750295766,
0.79449159519108975310913803326, 1.4180716080160267034452153921, 2.497596881362628499628013261231, 3.488722018641129583541073440628, 3.77875133380976331439897907119, 4.73474823543425807307530152230, 5.148256821172925656469287439129, 5.77154434739538243287819882496, 6.84796692285556999029992926648, 7.804483726350539428496383131647, 8.33891145088643773887284160914, 9.35836690324135033079909538059, 9.85375062711228100346171763735, 10.5394241472802479450815060227, 11.37785395161819191829415413919, 11.71114001799475688126693645309, 12.86671325638104858701907781857, 13.30462082909305716834214268193, 13.96067291966988542671968183799, 14.53212819726024453819029361179, 15.15612610992233708304806457837, 15.936207302742519274427544943018, 16.43578316944851201685605762340, 17.153563327390158203805353572608, 17.66013766551700918138090364836