L(s) = 1 | + (−0.349 − 0.605i)5-s + (0.229 + 0.132i)11-s + (1.13 − 0.657i)13-s − 3.72·17-s − 0.441i·19-s + (−4.29 + 2.48i)23-s + (2.25 − 3.90i)25-s + (0.273 + 0.157i)29-s + (−4.85 + 2.80i)31-s + 0.702·37-s + (5.39 + 9.34i)41-s + (3.73 − 6.46i)43-s + (−3.50 + 6.06i)47-s + 9.83i·53-s − 0.185i·55-s + ⋯ |
L(s) = 1 | + (−0.156 − 0.270i)5-s + (0.0692 + 0.0399i)11-s + (0.315 − 0.182i)13-s − 0.904·17-s − 0.101i·19-s + (−0.896 + 0.517i)23-s + (0.451 − 0.781i)25-s + (0.0507 + 0.0292i)29-s + (−0.872 + 0.503i)31-s + 0.115·37-s + (0.842 + 1.45i)41-s + (0.569 − 0.985i)43-s + (−0.510 + 0.884i)47-s + 1.35i·53-s − 0.0250i·55-s + ⋯ |
Λ(s)=(=(5292s/2ΓC(s)L(s)(−0.294−0.955i)Λ(2−s)
Λ(s)=(=(5292s/2ΓC(s+1/2)L(s)(−0.294−0.955i)Λ(1−s)
Degree: |
2 |
Conductor: |
5292
= 22⋅33⋅72
|
Sign: |
−0.294−0.955i
|
Analytic conductor: |
42.2568 |
Root analytic conductor: |
6.50052 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ5292(4409,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 5292, ( :1/2), −0.294−0.955i)
|
Particular Values
L(1) |
≈ |
0.8570659781 |
L(21) |
≈ |
0.8570659781 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
| 7 | 1 |
good | 5 | 1+(0.349+0.605i)T+(−2.5+4.33i)T2 |
| 11 | 1+(−0.229−0.132i)T+(5.5+9.52i)T2 |
| 13 | 1+(−1.13+0.657i)T+(6.5−11.2i)T2 |
| 17 | 1+3.72T+17T2 |
| 19 | 1+0.441iT−19T2 |
| 23 | 1+(4.29−2.48i)T+(11.5−19.9i)T2 |
| 29 | 1+(−0.273−0.157i)T+(14.5+25.1i)T2 |
| 31 | 1+(4.85−2.80i)T+(15.5−26.8i)T2 |
| 37 | 1−0.702T+37T2 |
| 41 | 1+(−5.39−9.34i)T+(−20.5+35.5i)T2 |
| 43 | 1+(−3.73+6.46i)T+(−21.5−37.2i)T2 |
| 47 | 1+(3.50−6.06i)T+(−23.5−40.7i)T2 |
| 53 | 1−9.83iT−53T2 |
| 59 | 1+(6.73+11.6i)T+(−29.5+51.0i)T2 |
| 61 | 1+(4.89+2.82i)T+(30.5+52.8i)T2 |
| 67 | 1+(−2.97−5.14i)T+(−33.5+58.0i)T2 |
| 71 | 1−13.4iT−71T2 |
| 73 | 1−7.69iT−73T2 |
| 79 | 1+(0.698−1.20i)T+(−39.5−68.4i)T2 |
| 83 | 1+(−3.72+6.45i)T+(−41.5−71.8i)T2 |
| 89 | 1−11.1T+89T2 |
| 97 | 1+(9.18+5.30i)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
−8.328419823888493165082662650655, −7.81098924667462160557116861665, −6.94589111647104476627161246068, −6.25899852508193792989561213708, −5.55976542555005553238729682230, −4.62947828024429568207234890716, −4.07607352779368679631288943854, −3.10234401279720911036225616929, −2.17370950830499542087816745538, −1.10330467521976215848156246864,
0.23812833573828397108895137382, 1.67048777960493427815750758768, 2.52095945589610122102896736865, 3.54759231677134077010970095330, 4.18911120127577210863701577463, 5.03566198123390034501314731792, 5.94160293286285378833500828068, 6.51129837964575077371947131675, 7.30975693062598967191955149196, 7.893500308601105164948506812340