L(s) = 1 | + (0.320 − 0.986i)3-s + (−7.72 + 5.61i)5-s + (5.56 + 17.1i)7-s + (20.9 + 15.2i)9-s + (10.0 + 35.0i)11-s + (−13.3 − 9.68i)13-s + (3.05 + 9.41i)15-s + (64.6 − 46.9i)17-s + (−44.2 + 136. i)19-s + 18.6·21-s − 45.6·23-s + (−10.4 + 32.2i)25-s + (44.4 − 32.2i)27-s + (−45.4 − 140. i)29-s + (12.6 + 9.17i)31-s + ⋯ |
L(s) = 1 | + (0.0616 − 0.189i)3-s + (−0.690 + 0.501i)5-s + (0.300 + 0.924i)7-s + (0.776 + 0.564i)9-s + (0.274 + 0.961i)11-s + (−0.284 − 0.206i)13-s + (0.0526 + 0.162i)15-s + (0.921 − 0.669i)17-s + (−0.534 + 1.64i)19-s + 0.194·21-s − 0.413·23-s + (−0.0837 + 0.257i)25-s + (0.316 − 0.229i)27-s + (−0.291 − 0.896i)29-s + (0.0731 + 0.0531i)31-s + ⋯ |
Λ(s)=(=(88s/2ΓC(s)L(s)(0.320−0.947i)Λ(4−s)
Λ(s)=(=(88s/2ΓC(s+3/2)L(s)(0.320−0.947i)Λ(1−s)
Degree: |
2 |
Conductor: |
88
= 23⋅11
|
Sign: |
0.320−0.947i
|
Analytic conductor: |
5.19216 |
Root analytic conductor: |
2.27863 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ88(81,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 88, ( :3/2), 0.320−0.947i)
|
Particular Values
L(2) |
≈ |
1.11146+0.797433i |
L(21) |
≈ |
1.11146+0.797433i |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 11 | 1+(−10.0−35.0i)T |
good | 3 | 1+(−0.320+0.986i)T+(−21.8−15.8i)T2 |
| 5 | 1+(7.72−5.61i)T+(38.6−118.i)T2 |
| 7 | 1+(−5.56−17.1i)T+(−277.+201.i)T2 |
| 13 | 1+(13.3+9.68i)T+(678.+2.08e3i)T2 |
| 17 | 1+(−64.6+46.9i)T+(1.51e3−4.67e3i)T2 |
| 19 | 1+(44.2−136.i)T+(−5.54e3−4.03e3i)T2 |
| 23 | 1+45.6T+1.21e4T2 |
| 29 | 1+(45.4+140.i)T+(−1.97e4+1.43e4i)T2 |
| 31 | 1+(−12.6−9.17i)T+(9.20e3+2.83e4i)T2 |
| 37 | 1+(89.0+273.i)T+(−4.09e4+2.97e4i)T2 |
| 41 | 1+(−92.6+285.i)T+(−5.57e4−4.05e4i)T2 |
| 43 | 1−125.T+7.95e4T2 |
| 47 | 1+(9.79−30.1i)T+(−8.39e4−6.10e4i)T2 |
| 53 | 1+(−421.−306.i)T+(4.60e4+1.41e5i)T2 |
| 59 | 1+(70.9+218.i)T+(−1.66e5+1.20e5i)T2 |
| 61 | 1+(600.−436.i)T+(7.01e4−2.15e5i)T2 |
| 67 | 1−505.T+3.00e5T2 |
| 71 | 1+(−689.+500.i)T+(1.10e5−3.40e5i)T2 |
| 73 | 1+(−74.0−227.i)T+(−3.14e5+2.28e5i)T2 |
| 79 | 1+(−592.−430.i)T+(1.52e5+4.68e5i)T2 |
| 83 | 1+(476.−346.i)T+(1.76e5−5.43e5i)T2 |
| 89 | 1−663.T+7.04e5T2 |
| 97 | 1+(−1.10e3−801.i)T+(2.82e5+8.68e5i)T2 |
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
−14.04971802855791245037696824608, −12.46006583828401738468480739157, −12.03409501621509028252765340579, −10.61894685065999531509403799743, −9.558233600976919367126902527861, −7.981100697439189103698221514180, −7.25881705224679824438559409423, −5.57917599008207211060214571201, −4.01378462793041007743419663751, −2.09724432465332709856840835441,
0.884404961763795539274292700933, 3.64969807218502900969894236056, 4.69862251548027173648108812646, 6.59216736086487050842810710852, 7.81708269147632165910779010020, 8.934530276723199209495809154380, 10.23565643109568536564366702518, 11.30582785588609036753635064385, 12.38854354767880553454020057521, 13.43566945201900483762710771181