L(s) = 1 | + (−1.13 + 1.96i)3-s − 3.99i·5-s + (0.981 − 0.566i)7-s + (−1.06 − 1.84i)9-s + (−0.981 − 0.566i)11-s + (−3.59 − 0.266i)13-s + (7.84 + 4.52i)15-s + (−0.5 − 0.866i)17-s + (−3.39 + 1.96i)19-s + 2.56i·21-s + (−4.59 + 7.96i)23-s − 10.9·25-s − 1.96·27-s + (−2.02 + 3.51i)29-s − 9.05i·31-s + ⋯ |
L(s) = 1 | + (−0.654 + 1.13i)3-s − 1.78i·5-s + (0.371 − 0.214i)7-s + (−0.355 − 0.615i)9-s + (−0.295 − 0.170i)11-s + (−0.997 − 0.0740i)13-s + (2.02 + 1.16i)15-s + (−0.121 − 0.210i)17-s + (−0.779 + 0.450i)19-s + 0.560i·21-s + (−0.958 + 1.66i)23-s − 2.19·25-s − 0.377·27-s + (−0.376 + 0.652i)29-s − 1.62i·31-s + ⋯ |
Λ(s)=(=(832s/2ΓC(s)L(s)(−0.895+0.446i)Λ(2−s)
Λ(s)=(=(832s/2ΓC(s+1/2)L(s)(−0.895+0.446i)Λ(1−s)
Degree: |
2 |
Conductor: |
832
= 26⋅13
|
Sign: |
−0.895+0.446i
|
Analytic conductor: |
6.64355 |
Root analytic conductor: |
2.57750 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ832(641,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 832, ( :1/2), −0.895+0.446i)
|
Particular Values
L(1) |
≈ |
0.0488835−0.207677i |
L(21) |
≈ |
0.0488835−0.207677i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 13 | 1+(3.59+0.266i)T |
good | 3 | 1+(1.13−1.96i)T+(−1.5−2.59i)T2 |
| 5 | 1+3.99iT−5T2 |
| 7 | 1+(−0.981+0.566i)T+(3.5−6.06i)T2 |
| 11 | 1+(0.981+0.566i)T+(5.5+9.52i)T2 |
| 17 | 1+(0.5+0.866i)T+(−8.5+14.7i)T2 |
| 19 | 1+(3.39−1.96i)T+(9.5−16.4i)T2 |
| 23 | 1+(4.59−7.96i)T+(−11.5−19.9i)T2 |
| 29 | 1+(2.02−3.51i)T+(−14.5−25.1i)T2 |
| 31 | 1+9.05iT−31T2 |
| 37 | 1+(2.16+1.24i)T+(18.5+32.0i)T2 |
| 41 | 1+(2.42+1.39i)T+(20.5+35.5i)T2 |
| 43 | 1+(−3.24−5.62i)T+(−21.5+37.2i)T2 |
| 47 | 1+6.79iT−47T2 |
| 53 | 1−8.92T+53T2 |
| 59 | 1+(−7.77+4.49i)T+(29.5−51.0i)T2 |
| 61 | 1+(3.89+6.74i)T+(−30.5+52.8i)T2 |
| 67 | 1+(8.36+4.82i)T+(33.5+58.0i)T2 |
| 71 | 1+(6.86−3.96i)T+(35.5−61.4i)T2 |
| 73 | 1+9.29iT−73T2 |
| 79 | 1+12.9T+79T2 |
| 83 | 1−5.73iT−83T2 |
| 89 | 1+(−12.1−7.01i)T+(44.5+77.0i)T2 |
| 97 | 1+(7.70−4.44i)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
−9.713852152347112756907748688123, −9.307917451276004435246575169287, −8.234813512649756088111076142912, −7.53714439958119048920253532424, −5.80311143699301186345121947835, −5.29788308080208709186588246371, −4.52750773694219081416850962596, −3.87589369729132173357941324867, −1.82682994441042935999692026541, −0.10597681615621885245657765640,
2.03168274348223613501746682388, 2.73741550835237352039293393700, 4.27918425659581991965164717656, 5.63516114469575871167772920068, 6.50168567099011332860051071513, 6.99652198774928423043266685049, 7.65450836284052770372767564406, 8.690728356885653481513869023316, 10.25062276131043177052867371227, 10.46041913292995635121862841421