L(s) = 1 | + (1.70 − 1.42i)2-s + (−0.320 − 1.70i)3-s + (0.511 − 2.89i)4-s + (−2.97 − 2.44i)6-s + (−0.653 − 3.70i)7-s + (−1.04 − 1.81i)8-s + (−2.79 + 1.09i)9-s + (5.03 + 1.83i)11-s + (−5.09 + 0.0589i)12-s + (−4.69 − 3.93i)13-s + (−6.41 − 5.37i)14-s + (1.15 + 0.418i)16-s + (−1.40 + 2.43i)17-s + (−3.20 + 5.85i)18-s + (1.71 + 2.96i)19-s + ⋯ |
L(s) = 1 | + (1.20 − 1.01i)2-s + (−0.185 − 0.982i)3-s + (0.255 − 1.44i)4-s + (−1.21 − 0.996i)6-s + (−0.247 − 1.40i)7-s + (−0.370 − 0.641i)8-s + (−0.931 + 0.363i)9-s + (1.51 + 0.553i)11-s + (−1.47 + 0.0170i)12-s + (−1.30 − 1.09i)13-s + (−1.71 − 1.43i)14-s + (0.287 + 0.104i)16-s + (−0.340 + 0.589i)17-s + (−0.754 + 1.37i)18-s + (0.393 + 0.681i)19-s + ⋯ |
Λ(s)=(=(675s/2ΓC(s)L(s)(−0.995+0.0927i)Λ(2−s)
Λ(s)=(=(675s/2ΓC(s+1/2)L(s)(−0.995+0.0927i)Λ(1−s)
Degree: |
2 |
Conductor: |
675
= 33⋅52
|
Sign: |
−0.995+0.0927i
|
Analytic conductor: |
5.38990 |
Root analytic conductor: |
2.32161 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ675(76,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 675, ( :1/2), −0.995+0.0927i)
|
Particular Values
L(1) |
≈ |
0.116680−2.51140i |
L(21) |
≈ |
0.116680−2.51140i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(0.320+1.70i)T |
| 5 | 1 |
good | 2 | 1+(−1.70+1.42i)T+(0.347−1.96i)T2 |
| 7 | 1+(0.653+3.70i)T+(−6.57+2.39i)T2 |
| 11 | 1+(−5.03−1.83i)T+(8.42+7.07i)T2 |
| 13 | 1+(4.69+3.93i)T+(2.25+12.8i)T2 |
| 17 | 1+(1.40−2.43i)T+(−8.5−14.7i)T2 |
| 19 | 1+(−1.71−2.96i)T+(−9.5+16.4i)T2 |
| 23 | 1+(0.155−0.882i)T+(−21.6−7.86i)T2 |
| 29 | 1+(−1.61+1.35i)T+(5.03−28.5i)T2 |
| 31 | 1+(−0.576+3.26i)T+(−29.1−10.6i)T2 |
| 37 | 1+(−1.73+3.00i)T+(−18.5−32.0i)T2 |
| 41 | 1+(1.85+1.55i)T+(7.11+40.3i)T2 |
| 43 | 1+(6.42+2.33i)T+(32.9+27.6i)T2 |
| 47 | 1+(−0.626−3.55i)T+(−44.1+16.0i)T2 |
| 53 | 1−7.07T+53T2 |
| 59 | 1+(5.74−2.09i)T+(45.1−37.9i)T2 |
| 61 | 1+(−0.0966−0.548i)T+(−57.3+20.8i)T2 |
| 67 | 1+(−12.1−10.1i)T+(11.6+65.9i)T2 |
| 71 | 1+(−7.71+13.3i)T+(−35.5−61.4i)T2 |
| 73 | 1+(−6.47−11.2i)T+(−36.5+63.2i)T2 |
| 79 | 1+(−7.60+6.38i)T+(13.7−77.7i)T2 |
| 83 | 1+(−3.66+3.07i)T+(14.4−81.7i)T2 |
| 89 | 1+(1.45+2.51i)T+(−44.5+77.0i)T2 |
| 97 | 1+(5.99+2.18i)T+(74.3+62.3i)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
−10.37055184539396328520706756977, −9.661805692038784100276846032744, −8.071155303012069804588355302635, −7.24375790701141922350258555158, −6.41673511614254238735354432318, −5.36115020465969817642432266686, −4.26801388421407642140901737897, −3.45527524867496362371079869162, −2.15608076410624486826964841180, −0.986997655536987028811533819836,
2.68939643962615577487686389785, 3.74987916670377856179644142567, 4.79800611786821973902084651512, 5.28117760173947119133811424404, 6.44197961181818771252231759801, 6.78820922805597929433731328797, 8.380635737934760442420791639369, 9.234848912119284310574799006068, 9.696918530828936192685359830616, 11.30040238170812720096486583454