L(s) = 1 | − 2·3-s + 2·7-s + 2·9-s − 2·13-s − 2·17-s − 8·19-s − 4·21-s − 10·23-s − 6·27-s − 10·37-s + 4·39-s + 12·41-s + 6·43-s − 14·47-s + 2·49-s + 4·51-s − 2·53-s + 16·57-s − 8·59-s − 4·61-s + 4·63-s − 14·67-s + 20·69-s − 18·73-s − 16·79-s + 11·81-s − 10·83-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.755·7-s + 2/3·9-s − 0.554·13-s − 0.485·17-s − 1.83·19-s − 0.872·21-s − 2.08·23-s − 1.15·27-s − 1.64·37-s + 0.640·39-s + 1.87·41-s + 0.914·43-s − 2.04·47-s + 2/7·49-s + 0.560·51-s − 0.274·53-s + 2.11·57-s − 1.04·59-s − 0.512·61-s + 0.503·63-s − 1.71·67-s + 2.40·69-s − 2.10·73-s − 1.80·79-s + 11/9·81-s − 1.09·83-s + ⋯ |
Λ(s)=(=(2560000s/2ΓC(s)2L(s)Λ(2−s)
Λ(s)=(=(2560000s/2ΓC(s+1/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
2560000
= 212⋅54
|
Sign: |
1
|
Analytic conductor: |
163.227 |
Root analytic conductor: |
3.57436 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
2
|
Selberg data: |
(4, 2560000, ( :1/2,1/2), 1)
|
Particular Values
L(1) |
= |
0 |
L(21) |
= |
0 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 5 | | 1 |
good | 3 | C22 | 1+2T+2T2+2pT3+p2T4 |
| 7 | C22 | 1−2T+2T2−2pT3+p2T4 |
| 11 | C22 | 1+14T2+p2T4 |
| 13 | C2 | (1−4T+pT2)(1+6T+pT2) |
| 17 | C22 | 1+2T+2T2+2pT3+p2T4 |
| 19 | C2 | (1+4T+pT2)2 |
| 23 | C22 | 1+10T+50T2+10pT3+p2T4 |
| 29 | C22 | 1+6T2+p2T4 |
| 31 | C22 | 1−58T2+p2T4 |
| 37 | C2 | (1−2T+pT2)(1+12T+pT2) |
| 41 | C2 | (1−6T+pT2)2 |
| 43 | C22 | 1−6T+18T2−6pT3+p2T4 |
| 47 | C22 | 1+14T+98T2+14pT3+p2T4 |
| 53 | C22 | 1+2T+2T2+2pT3+p2T4 |
| 59 | C2 | (1+4T+pT2)2 |
| 61 | C2 | (1+2T+pT2)2 |
| 67 | C22 | 1+14T+98T2+14pT3+p2T4 |
| 71 | C22 | 1−106T2+p2T4 |
| 73 | C22 | 1+18T+162T2+18pT3+p2T4 |
| 79 | C2 | (1+8T+pT2)2 |
| 83 | C22 | 1+10T+50T2+10pT3+p2T4 |
| 89 | C2 | (1−pT2)2 |
| 97 | C22 | 1−6T+18T2−6pT3+p2T4 |
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L(s)=p∏ j=1∏4(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−9.025637204908989211419191621093, −8.948862844657013768046769506814, −8.406837679272210433957835242484, −7.889536571156655130389814405102, −7.63282397342565243643138374584, −7.28854716051763515581253346644, −6.55940039953861065185824892891, −6.39077459417847177487225285081, −5.86575297264700713333873134176, −5.67345537595415015490197509243, −5.11131520097096109500924912143, −4.51785488697542070280998210141, −4.24887415762888992609219846950, −4.05225121294251739274196101591, −3.10534604220966593671916779248, −2.48054409441270760871203712167, −1.78081875858630517063994731620, −1.56730681119818120057759854084, 0, 0,
1.56730681119818120057759854084, 1.78081875858630517063994731620, 2.48054409441270760871203712167, 3.10534604220966593671916779248, 4.05225121294251739274196101591, 4.24887415762888992609219846950, 4.51785488697542070280998210141, 5.11131520097096109500924912143, 5.67345537595415015490197509243, 5.86575297264700713333873134176, 6.39077459417847177487225285081, 6.55940039953861065185824892891, 7.28854716051763515581253346644, 7.63282397342565243643138374584, 7.889536571156655130389814405102, 8.406837679272210433957835242484, 8.948862844657013768046769506814, 9.025637204908989211419191621093