L(s) = 1 | + 8·2-s − 18·3-s + 48·4-s − 18·5-s − 144·6-s + 256·8-s + 243·9-s − 144·10-s + 2·11-s − 864·12-s − 288·13-s + 324·15-s + 1.28e3·16-s − 1.53e3·17-s + 1.94e3·18-s − 1.18e3·19-s − 864·20-s + 16·22-s + 3.39e3·23-s − 4.60e3·24-s − 1.30e3·25-s − 2.30e3·26-s − 2.91e3·27-s − 3.97e3·29-s + 2.59e3·30-s − 7.59e3·31-s + 6.14e3·32-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.321·5-s − 1.63·6-s + 1.41·8-s + 9-s − 0.455·10-s + 0.00498·11-s − 1.73·12-s − 0.472·13-s + 0.371·15-s + 5/4·16-s − 1.28·17-s + 1.41·18-s − 0.754·19-s − 0.482·20-s + 0.00704·22-s + 1.33·23-s − 1.63·24-s − 0.416·25-s − 0.668·26-s − 0.769·27-s − 0.877·29-s + 0.525·30-s − 1.41·31-s + 1.06·32-s + ⋯ |
Λ(s)=(=(86436s/2ΓC(s)2L(s)Λ(6−s)
Λ(s)=(=(86436s/2ΓC(s+5/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
86436
= 22⋅32⋅74
|
Sign: |
1
|
Analytic conductor: |
2223.39 |
Root analytic conductor: |
6.86679 |
Motivic weight: |
5 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
2
|
Selberg data: |
(4, 86436, ( :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 | C1 | (1+p2T)2 |
| 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
−10.94051470496383352451219154701, −10.62306998003108148428554252806, −9.810306264878780583963266579585, −9.603927862172053627915586273507, −8.553666883184945449629858081451, −8.362188324449009716368627233228, −7.29526775415216150857973534705, −7.14889230383366722223268502247, −6.50884808899763086003673909495, −6.35243652157073466088126387616, −5.31144498956551657011629354982, −5.24575952664842368545525495579, −4.71043719111555090532257570228, −4.06735968326835452851113059023, −3.56773574337593234013412674276, −2.85417082178007813404195012162, −1.88309949483789137823819999482, −1.52483767391626227590160960001, 0, 0,
1.52483767391626227590160960001, 1.88309949483789137823819999482, 2.85417082178007813404195012162, 3.56773574337593234013412674276, 4.06735968326835452851113059023, 4.71043719111555090532257570228, 5.24575952664842368545525495579, 5.31144498956551657011629354982, 6.35243652157073466088126387616, 6.50884808899763086003673909495, 7.14889230383366722223268502247, 7.29526775415216150857973534705, 8.362188324449009716368627233228, 8.553666883184945449629858081451, 9.603927862172053627915586273507, 9.810306264878780583963266579585, 10.62306998003108148428554252806, 10.94051470496383352451219154701