L(s) = 1 | + (1.87 + 0.695i)2-s − 2.63·3-s + (3.03 + 2.60i)4-s + (1.21 + 4.84i)5-s + (−4.93 − 1.83i)6-s − 12.3·7-s + (3.86 + 7.00i)8-s − 2.07·9-s + (−1.09 + 9.93i)10-s − 3.31i·11-s + (−7.97 − 6.86i)12-s − 4.53i·13-s + (−23.1 − 8.57i)14-s + (−3.19 − 12.7i)15-s + (2.38 + 15.8i)16-s + 12.6i·17-s + ⋯ |
L(s) = 1 | + (0.937 + 0.347i)2-s − 0.877·3-s + (0.757 + 0.652i)4-s + (0.243 + 0.969i)5-s + (−0.822 − 0.305i)6-s − 1.76·7-s + (0.483 + 0.875i)8-s − 0.230·9-s + (−0.109 + 0.993i)10-s − 0.301i·11-s + (−0.664 − 0.572i)12-s − 0.348i·13-s + (−1.65 − 0.612i)14-s + (−0.213 − 0.850i)15-s + (0.148 + 0.988i)16-s + 0.743i·17-s + ⋯ |
Λ(s)=(=(220s/2ΓC(s)L(s)(−0.893−0.448i)Λ(3−s)
Λ(s)=(=(220s/2ΓC(s+1)L(s)(−0.893−0.448i)Λ(1−s)
Degree: |
2 |
Conductor: |
220
= 22⋅5⋅11
|
Sign: |
−0.893−0.448i
|
Analytic conductor: |
5.99456 |
Root analytic conductor: |
2.44838 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ220(199,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 220, ( :1), −0.893−0.448i)
|
Particular Values
L(23) |
≈ |
0.280521+1.18448i |
L(21) |
≈ |
0.280521+1.18448i |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−1.87−0.695i)T |
| 5 | 1+(−1.21−4.84i)T |
| 11 | 1+3.31iT |
good | 3 | 1+2.63T+9T2 |
| 7 | 1+12.3T+49T2 |
| 13 | 1+4.53iT−169T2 |
| 17 | 1−12.6iT−289T2 |
| 19 | 1−16.3iT−361T2 |
| 23 | 1+11.8T+529T2 |
| 29 | 1−27.7T+841T2 |
| 31 | 1+2.38iT−961T2 |
| 37 | 1−59.7iT−1.36e3T2 |
| 41 | 1−16.4T+1.68e3T2 |
| 43 | 1+48.6T+1.84e3T2 |
| 47 | 1−80.3T+2.20e3T2 |
| 53 | 1+66.9iT−2.80e3T2 |
| 59 | 1+12.4iT−3.48e3T2 |
| 61 | 1−2.82T+3.72e3T2 |
| 67 | 1−55.3T+4.48e3T2 |
| 71 | 1−124.iT−5.04e3T2 |
| 73 | 1+130.iT−5.32e3T2 |
| 79 | 1−74.9iT−6.24e3T2 |
| 83 | 1−19.8T+6.88e3T2 |
| 89 | 1+92.5T+7.92e3T2 |
| 97 | 1−7.08iT−9.40e3T2 |
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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
−12.47394385704920289187487970028, −11.73101209196305741205139614819, −10.62435836336943140912590029662, −9.988971827957549988585778408485, −8.239637449578214888656107834518, −6.76609097743147624843492133569, −6.28039785831927344282644220517, −5.55440179457644862858658491135, −3.73633480113593235918570087791, −2.80676338962456764528767269621,
0.51782517297914865066370845432, 2.69446113142784496104007743739, 4.21706146216638167480431124142, 5.36015494365793337276421024625, 6.16848209881777056493048869180, 7.05238036402119427019980634856, 9.059816819260354340759114495312, 9.848501414528109502469473947651, 10.87891706376037814183669955190, 12.08799922237947939413343067930