L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (1.5 + 0.866i)5-s + (0.499 + 0.866i)6-s − 7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (1.5 − 0.866i)10-s + 0.999·12-s + (−1.5 + 0.866i)13-s + (−0.5 + 0.866i)14-s + (−1.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s − 0.999·18-s + (−0.5 + 0.866i)19-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (1.5 + 0.866i)5-s + (0.499 + 0.866i)6-s − 7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (1.5 − 0.866i)10-s + 0.999·12-s + (−1.5 + 0.866i)13-s + (−0.5 + 0.866i)14-s + (−1.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s − 0.999·18-s + (−0.5 + 0.866i)19-s + ⋯ |
Λ(s)=(=(3192s/2ΓC(s)L(s)(−0.0977−0.995i)Λ(1−s)
Λ(s)=(=(3192s/2ΓC(s)L(s)(−0.0977−0.995i)Λ(1−s)
Degree: |
2 |
Conductor: |
3192
= 23⋅3⋅7⋅19
|
Sign: |
−0.0977−0.995i
|
Analytic conductor: |
1.59301 |
Root analytic conductor: |
1.26214 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ3192(293,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 3192, ( :0), −0.0977−0.995i)
|
Particular Values
L(21) |
≈ |
0.8341476867 |
L(21) |
≈ |
0.8341476867 |
L(1) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.5+0.866i)T |
| 3 | 1+(0.5−0.866i)T |
| 7 | 1+T |
| 19 | 1+(0.5−0.866i)T |
good | 5 | 1+(−1.5−0.866i)T+(0.5+0.866i)T2 |
| 11 | 1+T2 |
| 13 | 1+(1.5−0.866i)T+(0.5−0.866i)T2 |
| 17 | 1+(−0.5−0.866i)T2 |
| 23 | 1+(1.5−0.866i)T+(0.5−0.866i)T2 |
| 29 | 1+(0.5−0.866i)T2 |
| 31 | 1+T2 |
| 37 | 1+T2 |
| 41 | 1+(0.5+0.866i)T2 |
| 43 | 1+(0.5+0.866i)T2 |
| 47 | 1+(−0.5+0.866i)T2 |
| 53 | 1+(0.5−0.866i)T2 |
| 59 | 1+(0.5−0.866i)T+(−0.5−0.866i)T2 |
| 61 | 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 |
| 67 | 1+(−0.5+0.866i)T2 |
| 71 | 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 |
| 73 | 1+(0.5+0.866i)T2 |
| 79 | 1+(0.5+0.866i)T2 |
| 83 | 1+1.73iT−T2 |
| 89 | 1+(0.5−0.866i)T2 |
| 97 | 1+(−0.5−0.866i)T2 |
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L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−9.651761870096336777678852755488, −8.857834216988361040129672692452, −7.23677279832285835682297373041, −6.30232489503217555916736865684, −5.99683365755137734581645831593, −5.25956555204983706503430516854, −4.29747229326049154964493192131, −3.48231288657808892139219937242, −2.61122308010183919382726456172, −1.86709166227319158341888201804,
0.40528198509220683181291794785, 2.18897756105296330703045774770, 2.78687064717018940671356240460, 4.35665576468890754379457018494, 5.23278290290801341763361311744, 5.59672456611350758674225639186, 6.46773586361200729411073917306, 6.76035619515725765372163340007, 7.81999388818909156422510957079, 8.445510600907515858122629948229