L(s) = 1 | + (0.994 + 0.104i)2-s + (−0.978 + 0.207i)3-s + (0.978 + 0.207i)4-s + (0.587 + 0.809i)5-s + (−0.994 + 0.104i)6-s + (0.207 − 0.978i)7-s + (0.951 + 0.309i)8-s + (0.913 − 0.406i)9-s + (0.5 + 0.866i)10-s − 12-s + (0.309 − 0.951i)14-s + (−0.743 − 0.669i)15-s + (0.913 + 0.406i)16-s + (0.104 + 0.994i)17-s + (0.951 − 0.309i)18-s + (0.743 − 0.669i)19-s + ⋯ |
L(s) = 1 | + (0.994 + 0.104i)2-s + (−0.978 + 0.207i)3-s + (0.978 + 0.207i)4-s + (0.587 + 0.809i)5-s + (−0.994 + 0.104i)6-s + (0.207 − 0.978i)7-s + (0.951 + 0.309i)8-s + (0.913 − 0.406i)9-s + (0.5 + 0.866i)10-s − 12-s + (0.309 − 0.951i)14-s + (−0.743 − 0.669i)15-s + (0.913 + 0.406i)16-s + (0.104 + 0.994i)17-s + (0.951 − 0.309i)18-s + (0.743 − 0.669i)19-s + ⋯ |
Λ(s)=(=(143s/2ΓR(s+1)L(s)(0.738+0.674i)Λ(1−s)
Λ(s)=(=(143s/2ΓR(s+1)L(s)(0.738+0.674i)Λ(1−s)
Degree: |
1 |
Conductor: |
143
= 11⋅13
|
Sign: |
0.738+0.674i
|
Analytic conductor: |
15.3674 |
Root analytic conductor: |
15.3674 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ143(97,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 143, (1: ), 0.738+0.674i)
|
Particular Values
L(21) |
≈ |
2.813910013+1.091710121i |
L(21) |
≈ |
2.813910013+1.091710121i |
L(1) |
≈ |
1.759511573+0.3947830909i |
L(1) |
≈ |
1.759511573+0.3947830909i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 11 | 1 |
| 13 | 1 |
good | 2 | 1+(0.994+0.104i)T |
| 3 | 1+(−0.978+0.207i)T |
| 5 | 1+(0.587+0.809i)T |
| 7 | 1+(0.207−0.978i)T |
| 17 | 1+(0.104+0.994i)T |
| 19 | 1+(0.743−0.669i)T |
| 23 | 1+(0.5+0.866i)T |
| 29 | 1+(0.669−0.743i)T |
| 31 | 1+(−0.587+0.809i)T |
| 37 | 1+(0.743+0.669i)T |
| 41 | 1+(0.207+0.978i)T |
| 43 | 1+(0.5−0.866i)T |
| 47 | 1+(0.951+0.309i)T |
| 53 | 1+(−0.809−0.587i)T |
| 59 | 1+(0.207−0.978i)T |
| 61 | 1+(−0.104−0.994i)T |
| 67 | 1+(−0.866+0.5i)T |
| 71 | 1+(−0.994+0.104i)T |
| 73 | 1+(−0.951+0.309i)T |
| 79 | 1+(−0.809−0.587i)T |
| 83 | 1+(0.587+0.809i)T |
| 89 | 1+(0.866−0.5i)T |
| 97 | 1+(−0.406−0.913i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−28.384885560905861965354957443960, −27.32486287755151949426994647280, −25.36411241220747860864255896530, −24.71510010524090179006578409080, −24.03621639576631307439046368222, −22.85429390981516610940611719964, −22.07686129158514184611108214200, −21.21170917562736916556226163456, −20.37746003032321998841206643451, −18.821191422773974879298389989527, −17.826778350642400069798818581185, −16.503802341876368479743753580273, −15.96543479787112893079297146249, −14.55064002083596771559869340350, −13.3518799576450552892759043586, −12.39219893275750038488934274012, −11.82497541344986201369084723087, −10.57706651187429010634608291733, −9.24012721068998988283728078823, −7.501105054509121241395031101485, −6.07591334873422168780469149536, −5.40839235552381417860518150139, −4.5011729182471660793539226319, −2.49364309515557037523257631501, −1.14019727113423447680612225417,
1.43515472526922232012110088622, 3.2652584618381412145711541997, 4.490338276698517743390152211988, 5.66420450696413274019439179437, 6.6482526539004176562020316145, 7.506952315793692336428765229428, 9.91053400691672550957345276420, 10.80521892641502614535836123120, 11.51805744689085287261470427253, 12.91319278427998043252768700738, 13.7974675109795681750506907898, 14.871920668551767341281340232787, 15.91254152159014000800191115711, 17.10733907340604009956951248494, 17.69180239560154663224098076187, 19.25377332857024361884299402433, 20.58552900373403838969605977079, 21.6286134987394452697026968303, 22.171712130432159162009988511309, 23.30505504111596136600625820185, 23.730239857630746218258139040752, 25.04376393143963099120312649955, 26.19495096474846081158753984339, 27.05104518131460714091452745782, 28.58162145545574193312295251266