L(s) = 1 | − i·2-s − 4-s + (−0.707 − 0.707i)5-s + 1.41·7-s + i·8-s + (−0.707 + 0.707i)10-s + (0.707 + 0.707i)11-s + (−0.707 + 0.707i)13-s − 1.41i·14-s + 16-s + 17-s + (1 + i)19-s + (0.707 + 0.707i)20-s + (0.707 − 0.707i)22-s − i·23-s + ⋯ |
L(s) = 1 | − i·2-s − 4-s + (−0.707 − 0.707i)5-s + 1.41·7-s + i·8-s + (−0.707 + 0.707i)10-s + (0.707 + 0.707i)11-s + (−0.707 + 0.707i)13-s − 1.41i·14-s + 16-s + 17-s + (1 + i)19-s + (0.707 + 0.707i)20-s + (0.707 − 0.707i)22-s − i·23-s + ⋯ |
Λ(s)=(=(2160s/2ΓC(s)L(s)(0.382+0.923i)Λ(1−s)
Λ(s)=(=(2160s/2ΓC(s)L(s)(0.382+0.923i)Λ(1−s)
Degree: |
2 |
Conductor: |
2160
= 24⋅33⋅5
|
Sign: |
0.382+0.923i
|
Analytic conductor: |
1.07798 |
Root analytic conductor: |
1.03825 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ2160(269,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 2160, ( :0), 0.382+0.923i)
|
Particular Values
L(21) |
≈ |
1.145510677 |
L(21) |
≈ |
1.145510677 |
L(1) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+iT |
| 3 | 1 |
| 5 | 1+(0.707+0.707i)T |
good | 7 | 1−1.41T+T2 |
| 11 | 1+(−0.707−0.707i)T+iT2 |
| 13 | 1+(0.707−0.707i)T−iT2 |
| 17 | 1−T+T2 |
| 19 | 1+(−1−i)T+iT2 |
| 23 | 1+iT−T2 |
| 29 | 1+(0.707−0.707i)T−iT2 |
| 31 | 1−T+T2 |
| 37 | 1+iT2 |
| 41 | 1+T2 |
| 43 | 1+(0.707+0.707i)T+iT2 |
| 47 | 1+T+T2 |
| 53 | 1+(−1+i)T−iT2 |
| 59 | 1+iT2 |
| 61 | 1+iT2 |
| 67 | 1−iT2 |
| 71 | 1+1.41T+T2 |
| 73 | 1+T2 |
| 79 | 1−T+T2 |
| 83 | 1+iT2 |
| 89 | 1−1.41T+T2 |
| 97 | 1+1.41iT−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
−9.208712827412928105776574307956, −8.357204361920694281803123970730, −7.893110472461591857894419402662, −7.04612737900362720761430417232, −5.48244397069680638827321513553, −4.85402560668873980919629450803, −4.26955174502167684606224047712, −3.40287507272241670787520641834, −1.94436799820489478805099549287, −1.23837974652002235132161079893,
1.08810651104447672285511050984, 2.94775284821295146102062262321, 3.81055102782744502031573539565, 4.80282883277623440770059015313, 5.42331498553022058175997726393, 6.34033660299625330191893121681, 7.34909997820923131555634725469, 7.74758212166572025151152236479, 8.270407491189313626747173264076, 9.242832497347617125344601510339