L(s) = 1 | − 3-s − 3·4-s − 4·7-s + 9-s + 3·12-s − 8·13-s + 5·16-s − 2·19-s + 4·21-s + 25-s − 27-s + 12·28-s − 3·36-s + 8·39-s − 20·43-s − 5·48-s − 2·49-s + 24·52-s + 2·57-s + 4·61-s − 4·63-s − 3·64-s − 32·67-s − 4·73-s − 75-s + 6·76-s − 16·79-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 3/2·4-s − 1.51·7-s + 1/3·9-s + 0.866·12-s − 2.21·13-s + 5/4·16-s − 0.458·19-s + 0.872·21-s + 1/5·25-s − 0.192·27-s + 2.26·28-s − 1/2·36-s + 1.28·39-s − 3.04·43-s − 0.721·48-s − 2/7·49-s + 3.32·52-s + 0.264·57-s + 0.512·61-s − 0.503·63-s − 3/8·64-s − 3.90·67-s − 0.468·73-s − 0.115·75-s + 0.688·76-s − 1.80·79-s + ⋯ |
Λ(s)=(=(243675s/2ΓC(s)2L(s)Λ(2−s)
Λ(s)=(=(243675s/2ΓC(s+1/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
243675
= 33⋅52⋅192
|
Sign: |
1
|
Analytic conductor: |
15.5369 |
Root analytic conductor: |
1.98536 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
2
|
Selberg data: |
(4, 243675, ( :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 | 3 | C1 | 1+T |
| 5 | C1×C1 | (1−T)(1+T) |
| 19 | C1 | (1+T)2 |
good | 2 | C2 | (1−T+pT2)(1+T+pT2) |
| 7 | C2 | (1+2T+pT2)2 |
| 11 | C2 | (1−2T+pT2)(1+2T+pT2) |
| 13 | C2 | (1+4T+pT2)2 |
| 17 | C2 | (1−2T+pT2)(1+2T+pT2) |
| 23 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 29 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 31 | C2 | (1+pT2)2 |
| 37 | C2 | (1+pT2)2 |
| 41 | C2 | (1+pT2)2 |
| 43 | C2 | (1+10T+pT2)2 |
| 47 | C2 | (1−12T+pT2)(1+12T+pT2) |
| 53 | C2 | (1−2T+pT2)(1+2T+pT2) |
| 59 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 61 | C2 | (1−2T+pT2)2 |
| 67 | C2 | (1+16T+pT2)2 |
| 71 | C2 | (1+pT2)2 |
| 73 | C2 | (1+2T+pT2)2 |
| 79 | C2 | (1+8T+pT2)2 |
| 83 | C2 | (1−12T+pT2)(1+12T+pT2) |
| 89 | C2 | (1+pT2)2 |
| 97 | C2 | (1+16T+pT2)2 |
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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
−8.450551892023889738347528214751, −8.255821268467971823002210114232, −7.28034875915901512150776673175, −7.20655519160802512316474001684, −6.50028762370085906668100412134, −6.11724584120824685520629468061, −5.26580762429388022199308033556, −5.15467699824668629469430799137, −4.39966598748280163950451872807, −4.19187417186236300755390321977, −3.13810729229125509539196623139, −2.93269055521134332794719136670, −1.68584050985555553185267943448, 0, 0,
1.68584050985555553185267943448, 2.93269055521134332794719136670, 3.13810729229125509539196623139, 4.19187417186236300755390321977, 4.39966598748280163950451872807, 5.15467699824668629469430799137, 5.26580762429388022199308033556, 6.11724584120824685520629468061, 6.50028762370085906668100412134, 7.20655519160802512316474001684, 7.28034875915901512150776673175, 8.255821268467971823002210114232, 8.450551892023889738347528214751