L(s) = 1 | + (−1.04 + 2.62i)2-s + 3i·3-s + (−5.81 − 5.49i)4-s − 7.49·5-s + (−7.88 − 3.13i)6-s − 9.78i·7-s + (20.5 − 9.53i)8-s − 9·9-s + (7.83 − 19.6i)10-s + 72.0·11-s + (16.4 − 17.4i)12-s + (−46.8 + 2.05i)13-s + (25.7 + 10.2i)14-s − 22.4i·15-s + (3.59 + 63.8i)16-s + 14.0·17-s + ⋯ |
L(s) = 1 | + (−0.369 + 0.929i)2-s + 0.577i·3-s + (−0.726 − 0.686i)4-s − 0.670·5-s + (−0.536 − 0.213i)6-s − 0.528i·7-s + (0.906 − 0.421i)8-s − 0.333·9-s + (0.247 − 0.622i)10-s + 1.97·11-s + (0.396 − 0.419i)12-s + (−0.999 + 0.0439i)13-s + (0.490 + 0.195i)14-s − 0.386i·15-s + (0.0561 + 0.998i)16-s + 0.200·17-s + ⋯ |
Λ(s)=(=(312s/2ΓC(s)L(s)(−0.460−0.887i)Λ(4−s)
Λ(s)=(=(312s/2ΓC(s+3/2)L(s)(−0.460−0.887i)Λ(1−s)
Degree: |
2 |
Conductor: |
312
= 23⋅3⋅13
|
Sign: |
−0.460−0.887i
|
Analytic conductor: |
18.4085 |
Root analytic conductor: |
4.29052 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ312(181,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 312, ( :3/2), −0.460−0.887i)
|
Particular Values
L(2) |
≈ |
1.184505123 |
L(21) |
≈ |
1.184505123 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(1.04−2.62i)T |
| 3 | 1−3iT |
| 13 | 1+(46.8−2.05i)T |
good | 5 | 1+7.49T+125T2 |
| 7 | 1+9.78iT−343T2 |
| 11 | 1−72.0T+1.33e3T2 |
| 17 | 1−14.0T+4.91e3T2 |
| 19 | 1−126.T+6.85e3T2 |
| 23 | 1+74.7T+1.21e4T2 |
| 29 | 1−177.iT−2.43e4T2 |
| 31 | 1−204.iT−2.97e4T2 |
| 37 | 1−262.T+5.06e4T2 |
| 41 | 1+376.iT−6.89e4T2 |
| 43 | 1−179.iT−7.95e4T2 |
| 47 | 1−4.21iT−1.03e5T2 |
| 53 | 1−102.iT−1.48e5T2 |
| 59 | 1−231.T+2.05e5T2 |
| 61 | 1−876.iT−2.26e5T2 |
| 67 | 1+597.T+3.00e5T2 |
| 71 | 1−576.iT−3.57e5T2 |
| 73 | 1−657.iT−3.89e5T2 |
| 79 | 1+137.T+4.93e5T2 |
| 83 | 1−1.08e3T+5.71e5T2 |
| 89 | 1+269.iT−7.04e5T2 |
| 97 | 1−1.73e3iT−9.12e5T2 |
show more | |
show less | |
L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−11.53947567895673986035317535469, −10.30663427965901806444364974154, −9.513674458265010595060923729163, −8.791317198646860638801246243497, −7.54989052839127265544126281062, −6.94116694288333441326695923553, −5.65585945130952517219206984440, −4.45903637342246737480387136350, −3.65635492441444926701992801445, −1.06249392947142418816384505738,
0.64972694753019423033842158365, 2.03487049054812027590155232920, 3.41703682906410060581237309132, 4.47758480108955316600700040135, 6.03996142586831235402914977926, 7.42073823198605277973640575224, 8.072907015690593536248147135230, 9.340838262681964120025918721630, 9.741027373783726730419614075139, 11.45683992653048152170014871932