L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s − 8-s + 10-s − 11-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + 19-s + (0.5 + 0.866i)20-s + (−0.5 − 0.866i)22-s + (0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + (0.5 − 0.866i)29-s + (−0.5 − 0.866i)31-s + (0.5 − 0.866i)32-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s − 8-s + 10-s − 11-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + 19-s + (0.5 + 0.866i)20-s + (−0.5 − 0.866i)22-s + (0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + (0.5 − 0.866i)29-s + (−0.5 − 0.866i)31-s + (0.5 − 0.866i)32-s + ⋯ |
Λ(s)=(=(273s/2ΓR(s+1)L(s)(0.949−0.313i)Λ(1−s)
Λ(s)=(=(273s/2ΓR(s+1)L(s)(0.949−0.313i)Λ(1−s)
Degree: |
1 |
Conductor: |
273
= 3⋅7⋅13
|
Sign: |
0.949−0.313i
|
Analytic conductor: |
29.3379 |
Root analytic conductor: |
29.3379 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ273(263,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 273, (1: ), 0.949−0.313i)
|
Particular Values
L(21) |
≈ |
2.017587176−0.3243166239i |
L(21) |
≈ |
2.017587176−0.3243166239i |
L(1) |
≈ |
1.289774281+0.2679502958i |
L(1) |
≈ |
1.289774281+0.2679502958i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 7 | 1 |
| 13 | 1 |
good | 2 | 1+(0.5+0.866i)T |
| 5 | 1+(0.5−0.866i)T |
| 11 | 1−T |
| 17 | 1+(0.5−0.866i)T |
| 19 | 1+T |
| 23 | 1+(0.5+0.866i)T |
| 29 | 1+(0.5−0.866i)T |
| 31 | 1+(−0.5−0.866i)T |
| 37 | 1+(−0.5−0.866i)T |
| 41 | 1+(0.5−0.866i)T |
| 43 | 1+(−0.5−0.866i)T |
| 47 | 1+(0.5−0.866i)T |
| 53 | 1+(0.5+0.866i)T |
| 59 | 1+(0.5−0.866i)T |
| 61 | 1+T |
| 67 | 1+T |
| 71 | 1+(0.5+0.866i)T |
| 73 | 1+(−0.5−0.866i)T |
| 79 | 1+(−0.5+0.866i)T |
| 83 | 1−T |
| 89 | 1+(0.5+0.866i)T |
| 97 | 1+(−0.5−0.866i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−25.64364515394006450899777170470, −24.41541161924455810313027947335, −23.43525006106548931682881803609, −22.704903200672051094308952281, −21.77756437936940916389498294542, −21.15463802900406081593830575123, −20.190512267213735372947239156518, −19.103779332235864282844961119460, −18.36449001937715853108044869698, −17.66410065152273770905571800200, −16.12705106538839629490785729814, −14.93425869116529289396353828027, −14.28272266875898266471114534394, −13.29038961515711574550061887855, −12.472401458359419979552557803531, −11.24306025308677378891067830749, −10.45181904830047813945978789096, −9.78855288854244502346291959786, −8.4247576988669418364249818976, −6.96424022102329053271278623209, −5.82104844052629442830232108026, −4.90437164284411319078473114127, −3.38237082221654179663562589912, −2.640706507031475806735522923, −1.31371042208338518080498781270,
0.561228181057559454870140900195, 2.48820604007981776898641099339, 3.88097711710668738109649131729, 5.281681687982364240516852970265, 5.51985619421060801545667895513, 7.114710362548041414570212346734, 7.966269221971018760200565741175, 9.062362045714410660498606208606, 9.926622968982147873051304763057, 11.60652233759857883988648930689, 12.57427859351963709986139824941, 13.468019067814047927932652615627, 14.057942876379270550130317850326, 15.469197034638504781214324624459, 16.055461777544200165468713303173, 17.01171296610047562350797461250, 17.819871527794500529168436298850, 18.74706596428036786517672955160, 20.35794981290993775582765512766, 20.99679966717458726441263370464, 21.83772777423640125089398767611, 22.9469883928535006029096332139, 23.69703657099995594957036314169, 24.60308726351423793534755861938, 25.22052619566339048461617187344