L(s) = 1 | + (0.486 − 1.49i)3-s + (1.80 − 1.31i)5-s + (−2.66 + 0.866i)7-s + (5.27 + 3.83i)9-s + (8.69 − 6.73i)11-s + (11.7 − 16.1i)13-s + (−1.08 − 3.34i)15-s + (−6.11 − 8.41i)17-s + (−14.5 − 4.71i)19-s + 4.41i·21-s − 9.47·23-s + (1.54 − 4.75i)25-s + (19.7 − 14.3i)27-s + (29.9 − 9.72i)29-s + (8.91 + 6.48i)31-s + ⋯ |
L(s) = 1 | + (0.162 − 0.498i)3-s + (0.361 − 0.262i)5-s + (−0.380 + 0.123i)7-s + (0.586 + 0.425i)9-s + (0.790 − 0.612i)11-s + (0.903 − 1.24i)13-s + (−0.0725 − 0.223i)15-s + (−0.359 − 0.495i)17-s + (−0.763 − 0.248i)19-s + 0.210i·21-s − 0.412·23-s + (0.0618 − 0.190i)25-s + (0.732 − 0.531i)27-s + (1.03 − 0.335i)29-s + (0.287 + 0.209i)31-s + ⋯ |
Λ(s)=(=(220s/2ΓC(s)L(s)(0.555+0.831i)Λ(3−s)
Λ(s)=(=(220s/2ΓC(s+1)L(s)(0.555+0.831i)Λ(1−s)
Degree: |
2 |
Conductor: |
220
= 22⋅5⋅11
|
Sign: |
0.555+0.831i
|
Analytic conductor: |
5.99456 |
Root analytic conductor: |
2.44838 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ220(161,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 220, ( :1), 0.555+0.831i)
|
Particular Values
L(23) |
≈ |
1.56769−0.837831i |
L(21) |
≈ |
1.56769−0.837831i |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 5 | 1+(−1.80+1.31i)T |
| 11 | 1+(−8.69+6.73i)T |
good | 3 | 1+(−0.486+1.49i)T+(−7.28−5.29i)T2 |
| 7 | 1+(2.66−0.866i)T+(39.6−28.8i)T2 |
| 13 | 1+(−11.7+16.1i)T+(−52.2−160.i)T2 |
| 17 | 1+(6.11+8.41i)T+(−89.3+274.i)T2 |
| 19 | 1+(14.5+4.71i)T+(292.+212.i)T2 |
| 23 | 1+9.47T+529T2 |
| 29 | 1+(−29.9+9.72i)T+(680.−494.i)T2 |
| 31 | 1+(−8.91−6.48i)T+(296.+913.i)T2 |
| 37 | 1+(−5.25−16.1i)T+(−1.10e3+804.i)T2 |
| 41 | 1+(−20.2−6.57i)T+(1.35e3+988.i)T2 |
| 43 | 1+31.6iT−1.84e3T2 |
| 47 | 1+(24.8−76.4i)T+(−1.78e3−1.29e3i)T2 |
| 53 | 1+(20.0+14.5i)T+(868.+2.67e3i)T2 |
| 59 | 1+(−17.3−53.5i)T+(−2.81e3+2.04e3i)T2 |
| 61 | 1+(−63.7−87.7i)T+(−1.14e3+3.53e3i)T2 |
| 67 | 1+46.4T+4.48e3T2 |
| 71 | 1+(100.−73.1i)T+(1.55e3−4.79e3i)T2 |
| 73 | 1+(−68.5+22.2i)T+(4.31e3−3.13e3i)T2 |
| 79 | 1+(13.8−19.0i)T+(−1.92e3−5.93e3i)T2 |
| 83 | 1+(53.1+73.1i)T+(−2.12e3+6.55e3i)T2 |
| 89 | 1+32.7T+7.92e3T2 |
| 97 | 1+(−141.−102.i)T+(2.90e3+8.94e3i)T2 |
show more | |
show less | |
L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−12.06533665902709290337209984801, −10.89137848299368771925774062810, −9.993160071179339568340435601913, −8.812600125998483508391382891019, −8.026265284960473161705390675380, −6.70277240320703914639663474206, −5.86628627248063063567155138939, −4.37376555881255639784126871618, −2.81342781870855316655188698518, −1.11981098770261820291371885842,
1.75962102442548852314490686115, 3.67415992569808480698762375598, 4.48242396732470774410814437199, 6.34167527255720959813925480615, 6.80548628651854089371832477898, 8.497201937083234351672761494717, 9.414766839561091579816116133069, 10.10962297353996702608535465411, 11.14610047969378062006552629711, 12.23074238837117137918573120281