L(s) = 1 | + (1.33 + 1.83i)3-s + (−3.12 − 3.90i)5-s + (−5.02 − 3.65i)7-s + (1.19 − 3.68i)9-s + (−6.42 − 8.92i)11-s + (5.44 − 16.7i)13-s + (2.99 − 10.9i)15-s + (9.49 + 29.2i)17-s + (−11.3 − 15.5i)19-s − 14.0i·21-s − 8.32i·23-s + (−5.50 + 24.3i)25-s + (27.7 − 9.00i)27-s + (22.9 − 31.5i)29-s + (−7.39 + 22.7i)31-s + ⋯ |
L(s) = 1 | + (0.443 + 0.610i)3-s + (−0.624 − 0.781i)5-s + (−0.718 − 0.522i)7-s + (0.133 − 0.409i)9-s + (−0.584 − 0.811i)11-s + (0.418 − 1.28i)13-s + (0.199 − 0.727i)15-s + (0.558 + 1.71i)17-s + (−0.596 − 0.820i)19-s − 0.670i·21-s − 0.362i·23-s + (−0.220 + 0.975i)25-s + (1.02 − 0.333i)27-s + (0.790 − 1.08i)29-s + (−0.238 + 0.734i)31-s + ⋯ |
Λ(s)=(=(220s/2ΓC(s)L(s)(0.0194+0.999i)Λ(3−s)
Λ(s)=(=(220s/2ΓC(s+1)L(s)(0.0194+0.999i)Λ(1−s)
Degree: |
2 |
Conductor: |
220
= 22⋅5⋅11
|
Sign: |
0.0194+0.999i
|
Analytic conductor: |
5.99456 |
Root analytic conductor: |
2.44838 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ220(189,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 220, ( :1), 0.0194+0.999i)
|
Particular Values
L(23) |
≈ |
0.835687−0.819601i |
L(21) |
≈ |
0.835687−0.819601i |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 5 | 1+(3.12+3.90i)T |
| 11 | 1+(6.42+8.92i)T |
good | 3 | 1+(−1.33−1.83i)T+(−2.78+8.55i)T2 |
| 7 | 1+(5.02+3.65i)T+(15.1+46.6i)T2 |
| 13 | 1+(−5.44+16.7i)T+(−136.−99.3i)T2 |
| 17 | 1+(−9.49−29.2i)T+(−233.+169.i)T2 |
| 19 | 1+(11.3+15.5i)T+(−111.+343.i)T2 |
| 23 | 1+8.32iT−529T2 |
| 29 | 1+(−22.9+31.5i)T+(−259.−799.i)T2 |
| 31 | 1+(7.39−22.7i)T+(−777.−564.i)T2 |
| 37 | 1+(12.3−16.9i)T+(−423.−1.30e3i)T2 |
| 41 | 1+(15.5+21.3i)T+(−519.+1.59e3i)T2 |
| 43 | 1+58.0T+1.84e3T2 |
| 47 | 1+(53.0+73.0i)T+(−682.+2.10e3i)T2 |
| 53 | 1+(−76.2−24.7i)T+(2.27e3+1.65e3i)T2 |
| 59 | 1+(−55.1−40.0i)T+(1.07e3+3.31e3i)T2 |
| 61 | 1+(−41.2+13.4i)T+(3.01e3−2.18e3i)T2 |
| 67 | 1−78.6iT−4.48e3T2 |
| 71 | 1+(−21.9−67.5i)T+(−4.07e3+2.96e3i)T2 |
| 73 | 1+(47.8+34.8i)T+(1.64e3+5.06e3i)T2 |
| 79 | 1+(−51.1−16.6i)T+(5.04e3+3.66e3i)T2 |
| 83 | 1+(−20.7−63.9i)T+(−5.57e3+4.04e3i)T2 |
| 89 | 1−166.T+7.92e3T2 |
| 97 | 1+(84.5+27.4i)T+(7.61e3+5.53e3i)T2 |
show more | |
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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
−11.91681301396373693637807732967, −10.51144232125251125525327569764, −10.07774695288841803989105694644, −8.485458243984675225323240730349, −8.373649991438213373560945797340, −6.69684836400045615340093746373, −5.41280979769854528323007372564, −3.98719814228007108986439935678, −3.26563238293686016543238975095, −0.60898337750792502386558023175,
2.12838414275815792247537063762, 3.28472824161627183558972096233, 4.84837149044965469477774290752, 6.51873241364309885139966683023, 7.23721067884943552844727501560, 8.130890577067849817169774739748, 9.366826357469607181609904061472, 10.31979918382304950742501116973, 11.51257722940730551409064872488, 12.27882437479974243972676828537