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
−9.022403998966948878076308245494, −8.554126223711287589987772092405, −8.424339009357722946925410567447, −8.300404256868600231453452760880, −7.43025383532482429028851935526, −7.19640006904961567452076639548, −6.76465136796237618407577454979, −6.32951940951293738519417828941, −5.89421474597741629612160493291, −5.40195254399666811455824754033, −4.51328996778309303297454757287, −4.46243602273108243297969635538, −3.38045832044536420736745568087, −3.30670619039744093512593138165, −2.47471824809626313682612031407, −1.95065718014466454113459179888, −1.46726652333813200311760642965, −0.875893954111922832017054695497, 0, 0,
0.875893954111922832017054695497, 1.46726652333813200311760642965, 1.95065718014466454113459179888, 2.47471824809626313682612031407, 3.30670619039744093512593138165, 3.38045832044536420736745568087, 4.46243602273108243297969635538, 4.51328996778309303297454757287, 5.40195254399666811455824754033, 5.89421474597741629612160493291, 6.32951940951293738519417828941, 6.76465136796237618407577454979, 7.19640006904961567452076639548, 7.43025383532482429028851935526, 8.300404256868600231453452760880, 8.424339009357722946925410567447, 8.554126223711287589987772092405, 9.022403998966948878076308245494