L(s) = 1 | + (0.866 + 0.5i)2-s + (0.366 + 1.36i)3-s + (−0.500 − 0.866i)4-s + (2 + i)5-s + (−0.366 + 1.36i)6-s + (−1 − 1.73i)7-s − 3i·8-s + (0.866 − 0.5i)9-s + (1.23 + 1.86i)10-s + (0.366 + 1.36i)11-s + (0.999 − i)12-s − 1.99i·14-s + (−0.633 + 3.09i)15-s + (0.500 − 0.866i)16-s + (1.36 + 0.366i)17-s + 18-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.211 + 0.788i)3-s + (−0.250 − 0.433i)4-s + (0.894 + 0.447i)5-s + (−0.149 + 0.557i)6-s + (−0.377 − 0.654i)7-s − 1.06i·8-s + (0.288 − 0.166i)9-s + (0.389 + 0.590i)10-s + (0.110 + 0.411i)11-s + (0.288 − 0.288i)12-s − 0.534i·14-s + (−0.163 + 0.799i)15-s + (0.125 − 0.216i)16-s + (0.331 + 0.0887i)17-s + 0.235·18-s + ⋯ |
Λ(s)=(=(845s/2ΓC(s)L(s)(0.839−0.543i)Λ(2−s)
Λ(s)=(=(845s/2ΓC(s+1/2)L(s)(0.839−0.543i)Λ(1−s)
Degree: |
2 |
Conductor: |
845
= 5⋅132
|
Sign: |
0.839−0.543i
|
Analytic conductor: |
6.74735 |
Root analytic conductor: |
2.59756 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ845(657,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 845, ( :1/2), 0.839−0.543i)
|
Particular Values
L(1) |
≈ |
2.45395+0.725496i |
L(21) |
≈ |
2.45395+0.725496i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1+(−2−i)T |
| 13 | 1 |
good | 2 | 1+(−0.866−0.5i)T+(1+1.73i)T2 |
| 3 | 1+(−0.366−1.36i)T+(−2.59+1.5i)T2 |
| 7 | 1+(1+1.73i)T+(−3.5+6.06i)T2 |
| 11 | 1+(−0.366−1.36i)T+(−9.52+5.5i)T2 |
| 17 | 1+(−1.36−0.366i)T+(14.7+8.5i)T2 |
| 19 | 1+(−6.83−1.83i)T+(16.4+9.5i)T2 |
| 23 | 1+(−4.09+1.09i)T+(19.9−11.5i)T2 |
| 29 | 1+(14.5+25.1i)T2 |
| 31 | 1+(5+5i)T+31iT2 |
| 37 | 1+(−18.5−32.0i)T2 |
| 41 | 1+(9.56−2.56i)T+(35.5−20.5i)T2 |
| 43 | 1+(−0.366+1.36i)T+(−37.2−21.5i)T2 |
| 47 | 1+6T+47T2 |
| 53 | 1+(−5+5i)T−53iT2 |
| 59 | 1+(2.56−9.56i)T+(−51.0−29.5i)T2 |
| 61 | 1+(−7−12.1i)T+(−30.5+52.8i)T2 |
| 67 | 1+(3.46+2i)T+(33.5+58.0i)T2 |
| 71 | 1+(0.366−1.36i)T+(−61.4−35.5i)T2 |
| 73 | 1+10iT−73T2 |
| 79 | 1+2iT−79T2 |
| 83 | 1+6T+83T2 |
| 89 | 1+(6.83−1.83i)T+(77.0−44.5i)T2 |
| 97 | 1+(1.73−i)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
−10.00330345689814115180173370181, −9.751782758006219794185375006116, −8.950738579201673970620707721325, −7.29592278342521876728442913755, −6.78316521649286718765728522410, −5.69882536936759048747690989938, −5.00717456959150086359738553808, −3.96010405808115085474465899347, −3.18805179587204126384863910430, −1.33048370382117695537351313446,
1.41289348664036284252441248091, 2.58920020913594767061340126830, 3.42986486708478238690131639231, 4.97547141097405801596249248058, 5.44371472571557917579264891623, 6.63636174915279763552136641533, 7.54259700374529407597249766858, 8.525896234428663455730808003331, 9.157043015547866654615894841581, 10.00860688311320762578626365856