L(s) = 1 | + (4.78 − 3.01i)2-s + (13.7 − 28.8i)4-s + 25i·5-s − 56.4·7-s + (−21.0 − 179. i)8-s + (75.4 + 119. i)10-s − 261. i·11-s − 720. i·13-s + (−270. + 170. i)14-s + (−643. − 796. i)16-s + 1.87e3·17-s + 1.99e3i·19-s + (721. + 344. i)20-s + (−787. − 1.24e3i)22-s − 2.57e3·23-s + ⋯ |
L(s) = 1 | + (0.845 − 0.533i)2-s + (0.431 − 0.902i)4-s + 0.447i·5-s − 0.435·7-s + (−0.116 − 0.993i)8-s + (0.238 + 0.378i)10-s − 0.650i·11-s − 1.18i·13-s + (−0.368 + 0.232i)14-s + (−0.628 − 0.778i)16-s + 1.57·17-s + 1.26i·19-s + (0.403 + 0.192i)20-s + (−0.346 − 0.550i)22-s − 1.01·23-s + ⋯ |
Λ(s)=(=(360s/2ΓC(s)L(s)(−0.993+0.116i)Λ(6−s)
Λ(s)=(=(360s/2ΓC(s+5/2)L(s)(−0.993+0.116i)Λ(1−s)
Degree: |
2 |
Conductor: |
360
= 23⋅32⋅5
|
Sign: |
−0.993+0.116i
|
Analytic conductor: |
57.7381 |
Root analytic conductor: |
7.59856 |
Motivic weight: |
5 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ360(181,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 360, ( :5/2), −0.993+0.116i)
|
Particular Values
L(3) |
≈ |
1.893789390 |
L(21) |
≈ |
1.893789390 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−4.78+3.01i)T |
| 3 | 1 |
| 5 | 1−25iT |
good | 7 | 1+56.4T+1.68e4T2 |
| 11 | 1+261.iT−1.61e5T2 |
| 13 | 1+720.iT−3.71e5T2 |
| 17 | 1−1.87e3T+1.41e6T2 |
| 19 | 1−1.99e3iT−2.47e6T2 |
| 23 | 1+2.57e3T+6.43e6T2 |
| 29 | 1−1.70e3iT−2.05e7T2 |
| 31 | 1+7.73e3T+2.86e7T2 |
| 37 | 1+1.22e4iT−6.93e7T2 |
| 41 | 1+1.49e4T+1.15e8T2 |
| 43 | 1+1.81e4iT−1.47e8T2 |
| 47 | 1+2.14e3T+2.29e8T2 |
| 53 | 1−1.60e3iT−4.18e8T2 |
| 59 | 1−2.68e3iT−7.14e8T2 |
| 61 | 1+4.45e4iT−8.44e8T2 |
| 67 | 1−1.24e4iT−1.35e9T2 |
| 71 | 1+8.18e3T+1.80e9T2 |
| 73 | 1+4.10e4T+2.07e9T2 |
| 79 | 1−4.63e4T+3.07e9T2 |
| 83 | 1−6.16e4iT−3.93e9T2 |
| 89 | 1+5.32e4T+5.58e9T2 |
| 97 | 1+3.92e4T+8.58e9T2 |
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show less | |
L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−10.31729815656893805297178151507, −9.707208163288533357170667149913, −8.195665420382237564243755648096, −7.17183890756217614105776189799, −5.85121668201411534462030908350, −5.49150911436203420454280454933, −3.67652534147793455630545132513, −3.25650353783891998038891973743, −1.77584305025505061400310945511, −0.32724985338002001614988813868,
1.71334156750915753822295096510, 3.12421158080268826447540289508, 4.26699966708305401168219599983, 5.13219339965880535620367483661, 6.24645887524989864403480775960, 7.11495678030438707807720204463, 8.041244713476131031367194288282, 9.147102279131487418021280779502, 10.02888605630489800997346391492, 11.49278957330199262281645114410