L(s) = 1 | + (1.5 − 0.866i)3-s + 5-s + (2.5 − 0.866i)7-s + (1.5 − 2.59i)9-s + 3·11-s + (−0.5 + 0.866i)13-s + (1.5 − 0.866i)15-s + (−1.5 + 2.59i)17-s + (2.5 + 4.33i)19-s + (3 − 3.46i)21-s − 23-s − 4·25-s − 5.19i·27-s + (−4.5 − 7.79i)29-s + (2 + 3.46i)31-s + ⋯ |
L(s) = 1 | + (0.866 − 0.499i)3-s + 0.447·5-s + (0.944 − 0.327i)7-s + (0.5 − 0.866i)9-s + 0.904·11-s + (−0.138 + 0.240i)13-s + (0.387 − 0.223i)15-s + (−0.363 + 0.630i)17-s + (0.573 + 0.993i)19-s + (0.654 − 0.755i)21-s − 0.208·23-s − 0.800·25-s − 0.999i·27-s + (−0.835 − 1.44i)29-s + (0.359 + 0.622i)31-s + ⋯ |
Λ(s)=(=(1008s/2ΓC(s)L(s)(0.823+0.566i)Λ(2−s)
Λ(s)=(=(1008s/2ΓC(s+1/2)L(s)(0.823+0.566i)Λ(1−s)
Degree: |
2 |
Conductor: |
1008
= 24⋅32⋅7
|
Sign: |
0.823+0.566i
|
Analytic conductor: |
8.04892 |
Root analytic conductor: |
2.83706 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1008(193,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1008, ( :1/2), 0.823+0.566i)
|
Particular Values
L(1) |
≈ |
2.661830726 |
L(21) |
≈ |
2.661830726 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1+(−1.5+0.866i)T |
| 7 | 1+(−2.5+0.866i)T |
good | 5 | 1−T+5T2 |
| 11 | 1−3T+11T2 |
| 13 | 1+(0.5−0.866i)T+(−6.5−11.2i)T2 |
| 17 | 1+(1.5−2.59i)T+(−8.5−14.7i)T2 |
| 19 | 1+(−2.5−4.33i)T+(−9.5+16.4i)T2 |
| 23 | 1+T+23T2 |
| 29 | 1+(4.5+7.79i)T+(−14.5+25.1i)T2 |
| 31 | 1+(−2−3.46i)T+(−15.5+26.8i)T2 |
| 37 | 1+(2.5+4.33i)T+(−18.5+32.0i)T2 |
| 41 | 1+(3.5−6.06i)T+(−20.5−35.5i)T2 |
| 43 | 1+(−1.5−2.59i)T+(−21.5+37.2i)T2 |
| 47 | 1+(−4+6.92i)T+(−23.5−40.7i)T2 |
| 53 | 1+(4.5−7.79i)T+(−26.5−45.8i)T2 |
| 59 | 1+(2+3.46i)T+(−29.5+51.0i)T2 |
| 61 | 1+(1−1.73i)T+(−30.5−52.8i)T2 |
| 67 | 1+(−6−10.3i)T+(−33.5+58.0i)T2 |
| 71 | 1+8T+71T2 |
| 73 | 1+(−6.5+11.2i)T+(−36.5−63.2i)T2 |
| 79 | 1+(−4+6.92i)T+(−39.5−68.4i)T2 |
| 83 | 1+(6.5+11.2i)T+(−41.5+71.8i)T2 |
| 89 | 1+(−4.5−7.79i)T+(−44.5+77.0i)T2 |
| 97 | 1+(−8.5−14.7i)T+(−48.5+84.0i)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
−9.718064825195812181491392346196, −9.040151919805813430038464962107, −8.118943662606817609951073246647, −7.58504368879306099563759295735, −6.56404412218236368100298542352, −5.74309446452027759966945853436, −4.35161592462145940197429539428, −3.63520668692261658897688127217, −2.13329657963466697794068030030, −1.39928190328830203230809695440,
1.59898483877940763721174448712, 2.60927833244559101138015319216, 3.76790532616847037004619472395, 4.79243623790019665119544464351, 5.49046248812668576999836412958, 6.84199170077041713333442427253, 7.64715858611448764716170142053, 8.572924509691368573195944604944, 9.217845918084275416428514865098, 9.779234120622260103825673264826