L(s) = 1 | + (0.802 − 0.597i)2-s + (0.286 − 0.957i)4-s + (0.448 − 0.893i)5-s + (0.230 − 0.973i)7-s + (−0.342 − 0.939i)8-s + (−0.173 − 0.984i)10-s + (0.448 + 0.893i)11-s + (−0.396 − 0.918i)14-s + (−0.835 − 0.549i)16-s + (0.173 + 0.984i)17-s + (−0.342 − 0.939i)19-s + (−0.727 − 0.686i)20-s + (0.893 + 0.448i)22-s + (0.973 − 0.230i)23-s + (−0.597 − 0.802i)25-s + ⋯ |
L(s) = 1 | + (0.802 − 0.597i)2-s + (0.286 − 0.957i)4-s + (0.448 − 0.893i)5-s + (0.230 − 0.973i)7-s + (−0.342 − 0.939i)8-s + (−0.173 − 0.984i)10-s + (0.448 + 0.893i)11-s + (−0.396 − 0.918i)14-s + (−0.835 − 0.549i)16-s + (0.173 + 0.984i)17-s + (−0.342 − 0.939i)19-s + (−0.727 − 0.686i)20-s + (0.893 + 0.448i)22-s + (0.973 − 0.230i)23-s + (−0.597 − 0.802i)25-s + ⋯ |
Λ(s)=(=(1053s/2ΓR(s)L(s)(−0.736−0.676i)Λ(1−s)
Λ(s)=(=(1053s/2ΓR(s)L(s)(−0.736−0.676i)Λ(1−s)
Degree: |
1 |
Conductor: |
1053
= 34⋅13
|
Sign: |
−0.736−0.676i
|
Analytic conductor: |
4.89011 |
Root analytic conductor: |
4.89011 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1053(527,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 1053, (0: ), −0.736−0.676i)
|
Particular Values
L(21) |
≈ |
0.9790659515−2.511901410i |
L(21) |
≈ |
0.9790659515−2.511901410i |
L(1) |
≈ |
1.385138891−1.182399946i |
L(1) |
≈ |
1.385138891−1.182399946i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 13 | 1 |
good | 2 | 1+(0.802−0.597i)T |
| 5 | 1+(0.448−0.893i)T |
| 7 | 1+(0.230−0.973i)T |
| 11 | 1+(0.448+0.893i)T |
| 17 | 1+(0.173+0.984i)T |
| 19 | 1+(−0.342−0.939i)T |
| 23 | 1+(0.973−0.230i)T |
| 29 | 1+(0.993−0.116i)T |
| 31 | 1+(0.727−0.686i)T |
| 37 | 1+(−0.642−0.766i)T |
| 41 | 1+(0.918−0.396i)T |
| 43 | 1+(−0.893+0.448i)T |
| 47 | 1+(−0.727−0.686i)T |
| 53 | 1+(0.5+0.866i)T |
| 59 | 1+(−0.448+0.893i)T |
| 61 | 1+(−0.686+0.727i)T |
| 67 | 1+(0.802+0.597i)T |
| 71 | 1+(−0.642−0.766i)T |
| 73 | 1+(0.342+0.939i)T |
| 79 | 1+(−0.993+0.116i)T |
| 83 | 1+(0.116+0.993i)T |
| 89 | 1+(−0.642+0.766i)T |
| 97 | 1+(−0.998−0.0581i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−21.66531067338668128570423475574, −21.49212928911119494748036118414, −20.64704651612040176928082825847, −19.30859563635234941444042478070, −18.6394279905738699463060271354, −17.84835515104551994641328906195, −17.07788015387117193051648108094, −16.14388286619786810400745380537, −15.461880073454907269500884071141, −14.58921770540827924226195132191, −14.145130235178116988539335613827, −13.381794435312755437058493601077, −12.32271244721817068283104274556, −11.62845888397070946184966918558, −10.936759188296409202267155565819, −9.74120781566093833722694973933, −8.7308693079846459572115101199, −8.02241892677912052230033431751, −6.86813154481890839768965379004, −6.298496472267303786632475276247, −5.50818497387619089087509834908, −4.7080567812914488529151565078, −3.25736101109060064961627061485, −2.94067481459664198358223634015, −1.71666003624758620610404992301,
0.887546274816796435285571007925, 1.6514821232340347034027790990, 2.70285352508473881318728403536, 4.0967157353681030794925785079, 4.46318699471230474258198378559, 5.35086774438372060662643738367, 6.41444290080486352319350798896, 7.16007739695325044749485639373, 8.43881886777758673114689070515, 9.39403292918626649883344189993, 10.168634155030225141756559688253, 10.85490295771922688620402435484, 11.865447899409406286453541870690, 12.63790155999632259366274278773, 13.24532988257995934160338177903, 13.930596947965371902451951758780, 14.80192251588081411110035465427, 15.51405005574973605631064780813, 16.64728784874085497260799030278, 17.27523925514427812120750020128, 18.00705133856539641756622221335, 19.484398819582723682941650458222, 19.68974387213581379083848694977, 20.60525207611455636034552325095, 21.15797321954953183803622072186