L(s) = 1 | − 8·2-s + 48·4-s + 18·5-s − 256·8-s − 144·10-s − 2·11-s − 288·13-s + 1.28e3·16-s + 1.53e3·17-s − 1.18e3·19-s + 864·20-s + 16·22-s − 3.39e3·23-s − 1.30e3·25-s + 2.30e3·26-s + 3.97e3·29-s − 7.59e3·31-s − 6.14e3·32-s − 1.22e4·34-s + 2.68e3·37-s + 9.50e3·38-s − 4.60e3·40-s + 3.66e4·41-s − 2.30e4·43-s − 96·44-s + 2.71e4·46-s + 864·47-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.321·5-s − 1.41·8-s − 0.455·10-s − 0.00498·11-s − 0.472·13-s + 5/4·16-s + 1.28·17-s − 0.754·19-s + 0.482·20-s + 0.00704·22-s − 1.33·23-s − 0.416·25-s + 0.668·26-s + 0.877·29-s − 1.41·31-s − 1.06·32-s − 1.81·34-s + 0.322·37-s + 1.06·38-s − 0.455·40-s + 3.40·41-s − 1.89·43-s − 0.00747·44-s + 1.88·46-s + 0.0570·47-s + ⋯ |
Λ(s)=(=(777924s/2ΓC(s)2L(s)Λ(6−s)
Λ(s)=(=(777924s/2ΓC(s+5/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
777924
= 22⋅34⋅74
|
Sign: |
1
|
Analytic conductor: |
20010.5 |
Root analytic conductor: |
11.8936 |
Motivic weight: |
5 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
2
|
Selberg data: |
(4, 777924, ( :5/2,5/2), 1)
|
Particular Values
L(3) |
= |
0 |
L(21) |
= |
0 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | C1 | (1+p2T)2 |
| 3 | | 1 |
| 7 | | 1 |
good | 5 | D4 | 1−18T+1626T2−18p5T3+p10T4 |
| 11 | D4 | 1+2T−59002T2+2p5T3+p10T4 |
| 13 | D4 | 1+288T+688042T2+288p5T3+p10T4 |
| 17 | D4 | 1−90pT+2366314T2−90p6T3+p10T4 |
| 19 | D4 | 1+1188T+4834534T2+1188p5T3+p10T4 |
| 23 | D4 | 1+3390T+6218086T2+3390p5T3+p10T4 |
| 29 | D4 | 1−3976T+31254662T2−3976p5T3+p10T4 |
| 31 | D4 | 1+7596T+61727326T2+7596p5T3+p10T4 |
| 37 | D4 | 1−2688T+126774470T2−2688p5T3+p10T4 |
| 41 | D4 | 1−36630T+559995322T2−36630p5T3+p10T4 |
| 43 | D4 | 1+23032T+329072262T2+23032p5T3+p10T4 |
| 47 | D4 | 1−864T+46643358T2−864p5T3+p10T4 |
| 53 | D4 | 1−32920T+983844566T2−32920p5T3+p10T4 |
| 59 | D4 | 1+26712T+697343334T2+26712p5T3+p10T4 |
| 61 | D4 | 1+20412T+1230091258T2+20412p5T3+p10T4 |
| 67 | D4 | 1+36172T+33148290pT2+36172p5T3+p10T4 |
| 71 | D4 | 1+73706T+3356433686T2+73706p5T3+p10T4 |
| 73 | D4 | 1−74772T+3993671602T2−74772p5T3+p10T4 |
| 79 | D4 | 1+23116T+6249589662T2+23116p5T3+p10T4 |
| 83 | D4 | 1−147816T+13316083030T2−147816p5T3+p10T4 |
| 89 | D4 | 1−164646T+17878567722T2−164646p5T3+p10T4 |
| 97 | D4 | 1+162036T+20528488258T2+162036p5T3+p10T4 |
show more | | |
show less | | |
L(s)=p∏ j=1∏4(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−9.110987749020565277903405329118, −8.991401496919358282146114694328, −8.220400334419701025294000155163, −8.002081889761395316410637999759, −7.55430958542966741602841448909, −7.37080987351475880579049186904, −6.57017348092223294919284127802, −6.35577450239813307539667345171, −5.66512933122514901580679591304, −5.63181366803601933671415933532, −4.77707166905828990550874416558, −4.16936035130082708753465584189, −3.65836644967760464957873419020, −3.04556915047309937487484153527, −2.28413930294448249988341959805, −2.22497586143914281034170014326, −1.30001818799649595666258386704, −1.04420409080615409248609403174, 0, 0,
1.04420409080615409248609403174, 1.30001818799649595666258386704, 2.22497586143914281034170014326, 2.28413930294448249988341959805, 3.04556915047309937487484153527, 3.65836644967760464957873419020, 4.16936035130082708753465584189, 4.77707166905828990550874416558, 5.63181366803601933671415933532, 5.66512933122514901580679591304, 6.35577450239813307539667345171, 6.57017348092223294919284127802, 7.37080987351475880579049186904, 7.55430958542966741602841448909, 8.002081889761395316410637999759, 8.220400334419701025294000155163, 8.991401496919358282146114694328, 9.110987749020565277903405329118