L(s) = 1 | + (−0.965 − 0.258i)2-s + (0.866 + 0.5i)4-s + (0.793 − 0.608i)5-s + (−0.793 − 0.608i)7-s + (−0.707 − 0.707i)8-s + (−0.923 + 0.382i)10-s + (0.608 − 0.793i)11-s + (−0.866 − 0.5i)13-s + (0.608 + 0.793i)14-s + (0.5 + 0.866i)16-s + (−0.707 + 0.707i)19-s + (0.991 − 0.130i)20-s + (−0.793 + 0.608i)22-s + (0.991 + 0.130i)23-s + (0.258 − 0.965i)25-s + (0.707 + 0.707i)26-s + ⋯ |
L(s) = 1 | + (−0.965 − 0.258i)2-s + (0.866 + 0.5i)4-s + (0.793 − 0.608i)5-s + (−0.793 − 0.608i)7-s + (−0.707 − 0.707i)8-s + (−0.923 + 0.382i)10-s + (0.608 − 0.793i)11-s + (−0.866 − 0.5i)13-s + (0.608 + 0.793i)14-s + (0.5 + 0.866i)16-s + (−0.707 + 0.707i)19-s + (0.991 − 0.130i)20-s + (−0.793 + 0.608i)22-s + (0.991 + 0.130i)23-s + (0.258 − 0.965i)25-s + (0.707 + 0.707i)26-s + ⋯ |
Λ(s)=(=(153s/2ΓR(s)L(s)(−0.0648−0.997i)Λ(1−s)
Λ(s)=(=(153s/2ΓR(s)L(s)(−0.0648−0.997i)Λ(1−s)
Degree: |
1 |
Conductor: |
153
= 32⋅17
|
Sign: |
−0.0648−0.997i
|
Analytic conductor: |
0.710529 |
Root analytic conductor: |
0.710529 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ153(29,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 153, (0: ), −0.0648−0.997i)
|
Particular Values
L(21) |
≈ |
0.4733416763−0.5051213206i |
L(21) |
≈ |
0.4733416763−0.5051213206i |
L(1) |
≈ |
0.6596726469−0.2829276906i |
L(1) |
≈ |
0.6596726469−0.2829276906i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 17 | 1 |
good | 2 | 1+(−0.965−0.258i)T |
| 5 | 1+(0.793−0.608i)T |
| 7 | 1+(−0.793−0.608i)T |
| 11 | 1+(0.608−0.793i)T |
| 13 | 1+(−0.866−0.5i)T |
| 19 | 1+(−0.707+0.707i)T |
| 23 | 1+(0.991+0.130i)T |
| 29 | 1+(−0.130−0.991i)T |
| 31 | 1+(−0.608−0.793i)T |
| 37 | 1+(0.382−0.923i)T |
| 41 | 1+(0.130−0.991i)T |
| 43 | 1+(−0.258+0.965i)T |
| 47 | 1+(−0.866+0.5i)T |
| 53 | 1+(0.707−0.707i)T |
| 59 | 1+(0.965−0.258i)T |
| 61 | 1+(0.793+0.608i)T |
| 67 | 1+(0.5−0.866i)T |
| 71 | 1+(−0.382+0.923i)T |
| 73 | 1+(0.923+0.382i)T |
| 79 | 1+(−0.608+0.793i)T |
| 83 | 1+(0.965+0.258i)T |
| 89 | 1+iT |
| 97 | 1+(0.130+0.991i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−28.2983363085972893509775927888, −27.200385587815728262454037794044, −26.18722086092441727568836060318, −25.429645775001970857526671905700, −24.870318629327544328735216781930, −23.49506079057024709871017649572, −22.23148107244289102336652517010, −21.442354724406955368030261173507, −20.02224009040918868005972034823, −19.18719905943312271566788944070, −18.297168996997269528193534018555, −17.33574168598487044280410531839, −16.53524921626397186218560010930, −15.13414144496788405279560956946, −14.57626689956884095318764582501, −12.948699273959704687923049743926, −11.74157395225671466758547883441, −10.466852647349191782918780286953, −9.55216234924711091447155296532, −8.87375898400083409346308302483, −7.00968472379296018586351823026, −6.60257492888652709222479672932, −5.17938135554434298838974155252, −2.91991761253357436684103787248, −1.86041267007391990992944467691,
0.80370347766917070517475400664, 2.372951156810953749663136824518, 3.816672108661990309520504864921, 5.748139167586603808830219737802, 6.80744940179248992482472272792, 8.12713939369360895666564762589, 9.308407713087548296676348212076, 9.967350712088731272093359335403, 11.09572788937731289259150138167, 12.48748954306872550471010266837, 13.26185309784672800277135524521, 14.718648877340411326043728513635, 16.246311335533586482720983648586, 16.88269529814398042552797143506, 17.57638673380463180680156604948, 18.99947121412886321260646990602, 19.67771565353946293742084862724, 20.71122343858037222049271880764, 21.57749867532898191591078031669, 22.690839280437872971721188381747, 24.27712266080538865360052098470, 25.00160104427069152600399508857, 25.85731311879367017827981578208, 26.88161242010420347316989669261, 27.67634494624341571894471782329