L(s) = 1 | + (0.965 + 0.258i)2-s + (0.866 − 0.5i)3-s + (0.866 + 0.499i)4-s + (2.50 + 2.50i)5-s + (0.965 − 0.258i)6-s + (−1.87 − 1.86i)7-s + (0.707 + 0.707i)8-s + (0.499 − 0.866i)9-s + (1.77 + 3.06i)10-s + (−0.883 + 3.29i)11-s + 12-s + (1.87 − 3.08i)13-s + (−1.32 − 2.28i)14-s + (3.41 + 0.916i)15-s + (0.500 + 0.866i)16-s + (−0.906 + 1.57i)17-s + ⋯ |
L(s) = 1 | + (0.683 + 0.183i)2-s + (0.499 − 0.288i)3-s + (0.433 + 0.249i)4-s + (1.11 + 1.11i)5-s + (0.394 − 0.105i)6-s + (−0.709 − 0.705i)7-s + (0.249 + 0.249i)8-s + (0.166 − 0.288i)9-s + (0.559 + 0.969i)10-s + (−0.266 + 0.993i)11-s + 0.288·12-s + (0.518 − 0.854i)13-s + (−0.355 − 0.611i)14-s + (0.882 + 0.236i)15-s + (0.125 + 0.216i)16-s + (−0.219 + 0.380i)17-s + ⋯ |
Λ(s)=(=(546s/2ΓC(s)L(s)(0.887−0.460i)Λ(2−s)
Λ(s)=(=(546s/2ΓC(s+1/2)L(s)(0.887−0.460i)Λ(1−s)
Degree: |
2 |
Conductor: |
546
= 2⋅3⋅7⋅13
|
Sign: |
0.887−0.460i
|
Analytic conductor: |
4.35983 |
Root analytic conductor: |
2.08802 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ546(223,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 546, ( :1/2), 0.887−0.460i)
|
Particular Values
L(1) |
≈ |
2.72322+0.664759i |
L(21) |
≈ |
2.72322+0.664759i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.965−0.258i)T |
| 3 | 1+(−0.866+0.5i)T |
| 7 | 1+(1.87+1.86i)T |
| 13 | 1+(−1.87+3.08i)T |
good | 5 | 1+(−2.50−2.50i)T+5iT2 |
| 11 | 1+(0.883−3.29i)T+(−9.52−5.5i)T2 |
| 17 | 1+(0.906−1.57i)T+(−8.5−14.7i)T2 |
| 19 | 1+(−6.44+1.72i)T+(16.4−9.5i)T2 |
| 23 | 1+(5.86−3.38i)T+(11.5−19.9i)T2 |
| 29 | 1+(2.24+3.87i)T+(−14.5+25.1i)T2 |
| 31 | 1+(6.69+6.69i)T+31iT2 |
| 37 | 1+(−0.544+2.03i)T+(−32.0−18.5i)T2 |
| 41 | 1+(1.43−5.35i)T+(−35.5−20.5i)T2 |
| 43 | 1+(2.23+1.28i)T+(21.5+37.2i)T2 |
| 47 | 1+(−2.09+2.09i)T−47iT2 |
| 53 | 1−12.2T+53T2 |
| 59 | 1+(−0.202−0.753i)T+(−51.0+29.5i)T2 |
| 61 | 1+(3.49+2.01i)T+(30.5+52.8i)T2 |
| 67 | 1+(11.6+3.13i)T+(58.0+33.5i)T2 |
| 71 | 1+(1.20+4.48i)T+(−61.4+35.5i)T2 |
| 73 | 1+(−4.59+4.59i)T−73iT2 |
| 79 | 1−8.32T+79T2 |
| 83 | 1+(10.8+10.8i)T+83iT2 |
| 89 | 1+(6.87+1.84i)T+(77.0+44.5i)T2 |
| 97 | 1+(−1.49+0.401i)T+(84.0−48.5i)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
−10.71925853176903160000809746875, −9.997346290600912569211371597114, −9.418557374564260978183682141890, −7.70128592579457937486466467114, −7.25669084453866307288231424364, −6.24917261075135170315303712148, −5.56904774413443644412325455171, −3.92264709062449569988600305156, −3.02777960833541359769789408250, −1.96379759232745674623012129776,
1.59379829682392479623482979330, 2.83897002556809528450419081902, 3.96023697865023473115282282827, 5.35628654830092438590245660845, 5.69127632321221425224722948128, 6.85826877138938558780540679562, 8.423744143211347630125075785127, 9.074087375768042296945535548290, 9.698067008246533692061644106251, 10.66117070810307229771494944844