L(s) = 1 | − 2·3-s + 4-s + 4·7-s + 9-s − 2·12-s + 4·13-s + 16-s + 4·19-s − 8·21-s + 25-s + 4·27-s + 4·28-s − 20·31-s + 36-s + 2·37-s − 8·39-s − 8·43-s − 2·48-s − 2·49-s + 4·52-s − 8·57-s − 20·61-s + 4·63-s + 64-s + 4·67-s + 4·73-s − 2·75-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1/2·4-s + 1.51·7-s + 1/3·9-s − 0.577·12-s + 1.10·13-s + 1/4·16-s + 0.917·19-s − 1.74·21-s + 1/5·25-s + 0.769·27-s + 0.755·28-s − 3.59·31-s + 1/6·36-s + 0.328·37-s − 1.28·39-s − 1.21·43-s − 0.288·48-s − 2/7·49-s + 0.554·52-s − 1.05·57-s − 2.56·61-s + 0.503·63-s + 1/8·64-s + 0.488·67-s + 0.468·73-s − 0.230·75-s + ⋯ |
Λ(s)=(=(1232100s/2ΓC(s)2L(s)−Λ(2−s)
Λ(s)=(=(1232100s/2ΓC(s+1/2)2L(s)−Λ(1−s)
Degree: |
4 |
Conductor: |
1232100
= 22⋅32⋅52⋅372
|
Sign: |
−1
|
Analytic conductor: |
78.5597 |
Root analytic conductor: |
2.97714 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
1
|
Selberg data: |
(4, 1232100, ( :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 | C1×C1 | (1−T)(1+T) |
| 3 | C2 | 1+2T+pT2 |
| 5 | C1×C1 | (1−T)(1+T) |
| 37 | C1 | (1−T)2 |
good | 7 | C2 | (1−2T+pT2)2 |
| 11 | C2 | (1+pT2)2 |
| 13 | C2 | (1−2T+pT2)2 |
| 17 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 19 | C2 | (1−2T+pT2)2 |
| 23 | C2 | (1+pT2)2 |
| 29 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 31 | C2 | (1+10T+pT2)2 |
| 41 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 43 | C2 | (1+4T+pT2)2 |
| 47 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 53 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 59 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 61 | C2 | (1+10T+pT2)2 |
| 67 | C2 | (1−2T+pT2)2 |
| 71 | C2 | (1+pT2)2 |
| 73 | C2 | (1−2T+pT2)2 |
| 79 | C2 | (1+10T+pT2)2 |
| 83 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 89 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 97 | C2 | (1−2T+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
−7.59731631879469280117277624063, −7.37364372891229977185623115606, −7.01389829611117685755868614566, −6.24527726803627967890931657857, −6.09945696108452207540402656878, −5.49324095653367495167136231203, −5.25330564043371456613299060211, −4.80762486474812590647702092154, −4.31005201030842057232591337712, −3.49870387163770249696581998319, −3.29880100176274410685427430205, −2.29716886023620970672769162796, −1.44505436450462837187908934432, −1.42055400977912848245367112580, 0,
1.42055400977912848245367112580, 1.44505436450462837187908934432, 2.29716886023620970672769162796, 3.29880100176274410685427430205, 3.49870387163770249696581998319, 4.31005201030842057232591337712, 4.80762486474812590647702092154, 5.25330564043371456613299060211, 5.49324095653367495167136231203, 6.09945696108452207540402656878, 6.24527726803627967890931657857, 7.01389829611117685755868614566, 7.37364372891229977185623115606, 7.59731631879469280117277624063