L(s) = 1 | + (1.91 + 3.32i)2-s + (2.23 + 1.99i)3-s + (−5.35 + 9.27i)4-s + (0.187 + 0.108i)5-s + (−2.34 + 11.2i)6-s − 25.7·8-s + (1.02 + 8.94i)9-s + 0.830i·10-s + (−2.22 − 3.85i)11-s + (−30.5 + 10.0i)12-s + (14.9 + 8.61i)13-s + (0.203 + 0.616i)15-s + (−27.9 − 48.4i)16-s + 5.38i·17-s + (−27.7 + 20.5i)18-s − 7.94i·19-s + ⋯ |
L(s) = 1 | + (0.959 + 1.66i)2-s + (0.746 + 0.665i)3-s + (−1.33 + 2.31i)4-s + (0.0375 + 0.0216i)5-s + (−0.390 + 1.87i)6-s − 3.21·8-s + (0.113 + 0.993i)9-s + 0.0830i·10-s + (−0.202 − 0.350i)11-s + (−2.54 + 0.839i)12-s + (1.14 + 0.663i)13-s + (0.0135 + 0.0411i)15-s + (−1.74 − 3.02i)16-s + 0.316i·17-s + (−1.54 + 1.14i)18-s − 0.418i·19-s + ⋯ |
Λ(s)=(=(441s/2ΓC(s)L(s)(−0.812+0.582i)Λ(3−s)
Λ(s)=(=(441s/2ΓC(s+1)L(s)(−0.812+0.582i)Λ(1−s)
Degree: |
2 |
Conductor: |
441
= 32⋅72
|
Sign: |
−0.812+0.582i
|
Analytic conductor: |
12.0163 |
Root analytic conductor: |
3.46646 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ441(391,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 441, ( :1), −0.812+0.582i)
|
Particular Values
L(23) |
≈ |
0.964974−3.00296i |
L(21) |
≈ |
0.964974−3.00296i |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(−2.23−1.99i)T |
| 7 | 1 |
good | 2 | 1+(−1.91−3.32i)T+(−2+3.46i)T2 |
| 5 | 1+(−0.187−0.108i)T+(12.5+21.6i)T2 |
| 11 | 1+(2.22+3.85i)T+(−60.5+104.i)T2 |
| 13 | 1+(−14.9−8.61i)T+(84.5+146.i)T2 |
| 17 | 1−5.38iT−289T2 |
| 19 | 1+7.94iT−361T2 |
| 23 | 1+(−9.53+16.5i)T+(−264.5−458.i)T2 |
| 29 | 1+(−3.57−6.18i)T+(−420.5+728.i)T2 |
| 31 | 1+(−20.0−11.5i)T+(480.5+832.i)T2 |
| 37 | 1+10.3T+1.36e3T2 |
| 41 | 1+(−3.32−1.91i)T+(840.5+1.45e3i)T2 |
| 43 | 1+(−30.1−52.1i)T+(−924.5+1.60e3i)T2 |
| 47 | 1+(−42.4+24.4i)T+(1.10e3−1.91e3i)T2 |
| 53 | 1+50.0T+2.80e3T2 |
| 59 | 1+(−75.2−43.4i)T+(1.74e3+3.01e3i)T2 |
| 61 | 1+(35.6−20.5i)T+(1.86e3−3.22e3i)T2 |
| 67 | 1+(32.1−55.7i)T+(−2.24e3−3.88e3i)T2 |
| 71 | 1+11.8T+5.04e3T2 |
| 73 | 1−32.0iT−5.32e3T2 |
| 79 | 1+(7.26+12.5i)T+(−3.12e3+5.40e3i)T2 |
| 83 | 1+(−94.9+54.8i)T+(3.44e3−5.96e3i)T2 |
| 89 | 1+128.iT−7.92e3T2 |
| 97 | 1+(−63.3+36.5i)T+(4.70e3−8.14e3i)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
−11.63194332139017017290164842671, −10.42275894541452624627334057343, −9.043873393771725745468951015722, −8.576554139105132811190317193732, −7.75669192143904737440201481884, −6.66640930640397380947761420078, −5.82448360690835027465285696614, −4.66968636655980956961334713212, −3.98049463597930158534705454164, −2.88223109475429604315914074166,
0.952870871289612051552688547398, 2.06280816068259840774806166255, 3.19460469489594074485230612922, 3.93360342593390187362007918905, 5.31565571609271954234584009191, 6.26064929821923125650381642569, 7.73891816488263827204872343560, 8.922271936107180335302758559097, 9.648869322508740757538308016618, 10.60084019209356519572629300092