L(s) = 1 | + (−4 − 6.92i)5-s + (−20 + 34.6i)11-s − 12·13-s + (−29 + 50.2i)17-s + (−13 − 22.5i)19-s + (−32 − 55.4i)23-s + (30.5 − 52.8i)25-s + 62·29-s + (−126 + 218. i)31-s + (−13 − 22.5i)37-s − 6·41-s + 416·43-s + (−198 − 342. i)47-s + (−225 + 389. i)53-s + 320·55-s + ⋯ |
L(s) = 1 | + (−0.357 − 0.619i)5-s + (−0.548 + 0.949i)11-s − 0.256·13-s + (−0.413 + 0.716i)17-s + (−0.156 − 0.271i)19-s + (−0.290 − 0.502i)23-s + (0.244 − 0.422i)25-s + 0.397·29-s + (−0.730 + 1.26i)31-s + (−0.0577 − 0.100i)37-s − 0.0228·41-s + 1.47·43-s + (−0.614 − 1.06i)47-s + (−0.583 + 1.01i)53-s + 0.784·55-s + ⋯ |
Λ(s)=(=(1764s/2ΓC(s)L(s)(0.605+0.795i)Λ(4−s)
Λ(s)=(=(1764s/2ΓC(s+3/2)L(s)(0.605+0.795i)Λ(1−s)
Degree: |
2 |
Conductor: |
1764
= 22⋅32⋅72
|
Sign: |
0.605+0.795i
|
Analytic conductor: |
104.079 |
Root analytic conductor: |
10.2019 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1764(361,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1764, ( :3/2), 0.605+0.795i)
|
Particular Values
L(2) |
≈ |
1.316514793 |
L(21) |
≈ |
1.316514793 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
| 7 | 1 |
good | 5 | 1+(4+6.92i)T+(−62.5+108.i)T2 |
| 11 | 1+(20−34.6i)T+(−665.5−1.15e3i)T2 |
| 13 | 1+12T+2.19e3T2 |
| 17 | 1+(29−50.2i)T+(−2.45e3−4.25e3i)T2 |
| 19 | 1+(13+22.5i)T+(−3.42e3+5.94e3i)T2 |
| 23 | 1+(32+55.4i)T+(−6.08e3+1.05e4i)T2 |
| 29 | 1−62T+2.43e4T2 |
| 31 | 1+(126−218.i)T+(−1.48e4−2.57e4i)T2 |
| 37 | 1+(13+22.5i)T+(−2.53e4+4.38e4i)T2 |
| 41 | 1+6T+6.89e4T2 |
| 43 | 1−416T+7.95e4T2 |
| 47 | 1+(198+342.i)T+(−5.19e4+8.99e4i)T2 |
| 53 | 1+(225−389.i)T+(−7.44e4−1.28e5i)T2 |
| 59 | 1+(−137+237.i)T+(−1.02e5−1.77e5i)T2 |
| 61 | 1+(−288−498.i)T+(−1.13e5+1.96e5i)T2 |
| 67 | 1+(−238+412.i)T+(−1.50e5−2.60e5i)T2 |
| 71 | 1−448T+3.57e5T2 |
| 73 | 1+(−79+136.i)T+(−1.94e5−3.36e5i)T2 |
| 79 | 1+(−468−810.i)T+(−2.46e5+4.26e5i)T2 |
| 83 | 1+530T+5.71e5T2 |
| 89 | 1+(195+337.i)T+(−3.52e5+6.10e5i)T2 |
| 97 | 1−214T+9.12e5T2 |
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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
−8.708742635729165716083432075491, −8.149202634123166378105677685223, −7.25059372457522655395115084363, −6.53547911758221244498087788488, −5.41894259085668671644672603423, −4.66937361502452860721835247861, −3.99438439144310755964852714049, −2.70212319534457241403283708049, −1.72746003118927468358898249699, −0.40899398155359271184388571442,
0.69162672813409337139823330084, 2.21778901966792836883252487132, 3.10214587634098671619135994730, 3.92430849165303129349854604060, 5.03051243112049013496082168126, 5.85540061850950779536574930553, 6.70060123993165709978008939452, 7.56403956505456797244465115002, 8.098082506055592239600883389068, 9.105236130410637616120455831463