L(s) = 1 | + (2.5 + 4.33i)5-s + (−16 + 27.7i)7-s + (18 − 31.1i)11-s + (5 + 8.66i)13-s + 78·17-s + 140·19-s + (−96 − 166. i)23-s + (−12.5 + 21.6i)25-s + (3 − 5.19i)29-s + (8 + 13.8i)31-s − 160·35-s − 34·37-s + (−195 − 337. i)41-s + (26 − 45.0i)43-s + (204 − 353. i)47-s + ⋯ |
L(s) = 1 | + (0.223 + 0.387i)5-s + (−0.863 + 1.49i)7-s + (0.493 − 0.854i)11-s + (0.106 + 0.184i)13-s + 1.11·17-s + 1.69·19-s + (−0.870 − 1.50i)23-s + (−0.100 + 0.173i)25-s + (0.0192 − 0.0332i)29-s + (0.0463 + 0.0802i)31-s − 0.772·35-s − 0.151·37-s + (−0.742 − 1.28i)41-s + (0.0922 − 0.159i)43-s + (0.633 − 1.09i)47-s + ⋯ |
Λ(s)=(=(1620s/2ΓC(s)L(s)(0.939−0.342i)Λ(4−s)
Λ(s)=(=(1620s/2ΓC(s+3/2)L(s)(0.939−0.342i)Λ(1−s)
Degree: |
2 |
Conductor: |
1620
= 22⋅34⋅5
|
Sign: |
0.939−0.342i
|
Analytic conductor: |
95.5830 |
Root analytic conductor: |
9.77666 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1620(541,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1620, ( :3/2), 0.939−0.342i)
|
Particular Values
L(2) |
≈ |
2.194391073 |
L(21) |
≈ |
2.194391073 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
| 5 | 1+(−2.5−4.33i)T |
good | 7 | 1+(16−27.7i)T+(−171.5−297.i)T2 |
| 11 | 1+(−18+31.1i)T+(−665.5−1.15e3i)T2 |
| 13 | 1+(−5−8.66i)T+(−1.09e3+1.90e3i)T2 |
| 17 | 1−78T+4.91e3T2 |
| 19 | 1−140T+6.85e3T2 |
| 23 | 1+(96+166.i)T+(−6.08e3+1.05e4i)T2 |
| 29 | 1+(−3+5.19i)T+(−1.21e4−2.11e4i)T2 |
| 31 | 1+(−8−13.8i)T+(−1.48e4+2.57e4i)T2 |
| 37 | 1+34T+5.06e4T2 |
| 41 | 1+(195+337.i)T+(−3.44e4+5.96e4i)T2 |
| 43 | 1+(−26+45.0i)T+(−3.97e4−6.88e4i)T2 |
| 47 | 1+(−204+353.i)T+(−5.19e4−8.99e4i)T2 |
| 53 | 1−114T+1.48e5T2 |
| 59 | 1+(−258−446.i)T+(−1.02e5+1.77e5i)T2 |
| 61 | 1+(−29+50.2i)T+(−1.13e5−1.96e5i)T2 |
| 67 | 1+(−446−772.i)T+(−1.50e5+2.60e5i)T2 |
| 71 | 1−120T+3.57e5T2 |
| 73 | 1+646T+3.89e5T2 |
| 79 | 1+(−584+1.01e3i)T+(−2.46e5−4.26e5i)T2 |
| 83 | 1+(366−633.i)T+(−2.85e5−4.95e5i)T2 |
| 89 | 1−1.59e3T+7.04e5T2 |
| 97 | 1+(97−168.i)T+(−4.56e5−7.90e5i)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
−8.957443219870005762100105087169, −8.559191282727349600772239756330, −7.41502357686374101558462391567, −6.49982454698800300044380698471, −5.80661186852532379381577915685, −5.30347347061979727225001769932, −3.73188987453581386795549364296, −3.04872974283581631192514055274, −2.16088284888887560299574468514, −0.69545023403847283485533183278,
0.78532214588856797542208634786, 1.54381115221371941539635474003, 3.23082141514231525811268273839, 3.78206363679394617051609795565, 4.81702235819604419058825297831, 5.73585567407966762357358433129, 6.66372111873311704582122268005, 7.50061521551121798022346313356, 7.87363911714823365878502758221, 9.355700014624795915165473210464