L(s) = 1 | − 2-s − 2·3-s − 4-s + 2·6-s + 7-s + 3·8-s + 3·9-s + 2·12-s − 14-s − 16-s − 3·18-s − 8·19-s − 2·21-s − 6·24-s − 6·25-s − 4·27-s − 28-s − 4·29-s − 5·32-s − 3·36-s + 12·37-s + 8·38-s + 2·42-s + 2·48-s + 49-s + 6·50-s + 12·53-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.816·6-s + 0.377·7-s + 1.06·8-s + 9-s + 0.577·12-s − 0.267·14-s − 1/4·16-s − 0.707·18-s − 1.83·19-s − 0.436·21-s − 1.22·24-s − 6/5·25-s − 0.769·27-s − 0.188·28-s − 0.742·29-s − 0.883·32-s − 1/2·36-s + 1.97·37-s + 1.29·38-s + 0.308·42-s + 0.288·48-s + 1/7·49-s + 0.848·50-s + 1.64·53-s + ⋯ |
Λ(s)=(=(49392s/2ΓC(s)2L(s)−Λ(2−s)
Λ(s)=(=(49392s/2ΓC(s+1/2)2L(s)−Λ(1−s)
Degree: |
4 |
Conductor: |
49392
= 24⋅32⋅73
|
Sign: |
−1
|
Analytic conductor: |
3.14927 |
Root analytic conductor: |
1.33214 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
1
|
Selberg data: |
(4, 49392, ( :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 | C2 | 1+T+pT2 |
| 3 | C1 | (1+T)2 |
| 7 | C1 | 1−T |
good | 5 | C2 | (1−2T+pT2)(1+2T+pT2) |
| 11 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 13 | C2 | (1−2T+pT2)(1+2T+pT2) |
| 17 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 19 | C2 | (1+4T+pT2)2 |
| 23 | C2 | (1+pT2)2 |
| 29 | C2 | (1+2T+pT2)2 |
| 31 | C2 | (1+pT2)2 |
| 37 | C2 | (1−6T+pT2)2 |
| 41 | C2 | (1−2T+pT2)(1+2T+pT2) |
| 43 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 47 | C2 | (1+pT2)2 |
| 53 | C2 | (1−6T+pT2)2 |
| 59 | C2 | (1+12T+pT2)2 |
| 61 | C2 | (1−2T+pT2)(1+2T+pT2) |
| 67 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 71 | C2 | (1+pT2)2 |
| 73 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 79 | C2 | (1−16T+pT2)(1+16T+pT2) |
| 83 | C2 | (1−12T+pT2)2 |
| 89 | C2 | (1−14T+pT2)(1+14T+pT2) |
| 97 | C2 | (1−18T+pT2)(1+18T+pT2) |
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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
−9.863214598051417092763914791206, −9.198895146278336601365818133221, −9.163045893798789171919999202805, −8.068098981215555715874277637796, −7.977686193886306753143677629185, −7.38325958034671995701748313853, −6.53484921474790582970192984243, −6.19643457342138342025418009798, −5.48510765590071063887274763802, −4.94701466488767178340177905762, −4.13981751462080886088964913424, −4.01671863219499060051377207353, −2.39234840548789475685588947322, −1.40786771277750203137263322761, 0,
1.40786771277750203137263322761, 2.39234840548789475685588947322, 4.01671863219499060051377207353, 4.13981751462080886088964913424, 4.94701466488767178340177905762, 5.48510765590071063887274763802, 6.19643457342138342025418009798, 6.53484921474790582970192984243, 7.38325958034671995701748313853, 7.977686193886306753143677629185, 8.068098981215555715874277637796, 9.163045893798789171919999202805, 9.198895146278336601365818133221, 9.863214598051417092763914791206