L(s) = 1 | + (−1.37 + 0.314i)2-s + (1.00 + 0.580i)3-s + (1.80 − 0.866i)4-s − 5-s + (−1.56 − 0.484i)6-s + (−2.74 + 1.58i)7-s + (−2.21 + 1.76i)8-s + (−0.826 − 1.43i)9-s + (1.37 − 0.314i)10-s + (1.23 − 2.13i)11-s + (2.31 + 0.175i)12-s + (−3.60 − 0.150i)13-s + (3.28 − 3.04i)14-s + (−1.00 − 0.580i)15-s + (2.49 − 3.12i)16-s + (0.369 + 0.639i)17-s + ⋯ |
L(s) = 1 | + (−0.975 + 0.222i)2-s + (0.580 + 0.335i)3-s + (0.901 − 0.433i)4-s − 0.447·5-s + (−0.640 − 0.197i)6-s + (−1.03 + 0.598i)7-s + (−0.782 + 0.622i)8-s + (−0.275 − 0.477i)9-s + (0.436 − 0.0993i)10-s + (0.371 − 0.643i)11-s + (0.668 + 0.0506i)12-s + (−0.999 − 0.0416i)13-s + (0.877 − 0.813i)14-s + (−0.259 − 0.149i)15-s + (0.624 − 0.780i)16-s + (0.0895 + 0.155i)17-s + ⋯ |
Λ(s)=(=(520s/2ΓC(s)L(s)(−0.393+0.919i)Λ(2−s)
Λ(s)=(=(520s/2ΓC(s+1/2)L(s)(−0.393+0.919i)Λ(1−s)
Degree: |
2 |
Conductor: |
520
= 23⋅5⋅13
|
Sign: |
−0.393+0.919i
|
Analytic conductor: |
4.15222 |
Root analytic conductor: |
2.03769 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ520(381,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 520, ( :1/2), −0.393+0.919i)
|
Particular Values
L(1) |
≈ |
0.188612−0.285840i |
L(21) |
≈ |
0.188612−0.285840i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(1.37−0.314i)T |
| 5 | 1+T |
| 13 | 1+(3.60+0.150i)T |
good | 3 | 1+(−1.00−0.580i)T+(1.5+2.59i)T2 |
| 7 | 1+(2.74−1.58i)T+(3.5−6.06i)T2 |
| 11 | 1+(−1.23+2.13i)T+(−5.5−9.52i)T2 |
| 17 | 1+(−0.369−0.639i)T+(−8.5+14.7i)T2 |
| 19 | 1+(4.31+7.47i)T+(−9.5+16.4i)T2 |
| 23 | 1+(−2.88+4.99i)T+(−11.5−19.9i)T2 |
| 29 | 1+(1.59+0.919i)T+(14.5+25.1i)T2 |
| 31 | 1−9.05iT−31T2 |
| 37 | 1+(1.35−2.35i)T+(−18.5−32.0i)T2 |
| 41 | 1+(−0.165−0.0952i)T+(20.5+35.5i)T2 |
| 43 | 1+(7.43−4.29i)T+(21.5−37.2i)T2 |
| 47 | 1+8.23iT−47T2 |
| 53 | 1+4.18iT−53T2 |
| 59 | 1+(5.72+9.91i)T+(−29.5+51.0i)T2 |
| 61 | 1+(−4.03+2.32i)T+(30.5−52.8i)T2 |
| 67 | 1+(−1.66+2.88i)T+(−33.5−58.0i)T2 |
| 71 | 1+(3.24−1.87i)T+(35.5−61.4i)T2 |
| 73 | 1−1.03iT−73T2 |
| 79 | 1+9.18T+79T2 |
| 83 | 1+1.79T+83T2 |
| 89 | 1+(−12.1−7.04i)T+(44.5+77.0i)T2 |
| 97 | 1+(14.8−8.56i)T+(48.5−84.0i)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
−10.34537044642805908278720904578, −9.453817875088727568639757195770, −8.843466854780396349829276609671, −8.311065559549145556198772687718, −6.83412052557288632239747765068, −6.46331723404372444381385200953, −4.99614085578644346882732364950, −3.32347536429935554109786659883, −2.57988806567746090943293200224, −0.23850039913141593039634511011,
1.81947966037899846799081034255, 3.03836663231841410039581612260, 4.08270555067138177090724052583, 5.90666351496517936681506198735, 7.15886690345862332931027653983, 7.51537772525961006027860131916, 8.459086991544363711731775308859, 9.490231794023385946913093569860, 10.04012227086343411267584201556, 10.96051751544935980408365514137