L(s) = 1 | − 4·7-s − 4·13-s − 12·19-s + 4·25-s + 8·31-s + 18·37-s + 12·43-s + 2·49-s + 8·61-s + 14·73-s − 8·79-s + 16·91-s − 10·97-s − 20·103-s − 18·109-s − 8·121-s + 127-s + 131-s + 48·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + ⋯ |
L(s) = 1 | − 1.51·7-s − 1.10·13-s − 2.75·19-s + 4/5·25-s + 1.43·31-s + 2.95·37-s + 1.82·43-s + 2/7·49-s + 1.02·61-s + 1.63·73-s − 0.900·79-s + 1.67·91-s − 1.01·97-s − 1.97·103-s − 1.72·109-s − 0.727·121-s + 0.0887·127-s + 0.0873·131-s + 4.16·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3/13·169-s + ⋯ |
Λ(s)=(=(876096s/2ΓC(s)2L(s)−Λ(2−s)
Λ(s)=(=(876096s/2ΓC(s+1/2)2L(s)−Λ(1−s)
Degree: |
4 |
Conductor: |
876096
= 26⋅34⋅132
|
Sign: |
−1
|
Analytic conductor: |
55.8606 |
Root analytic conductor: |
2.73386 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
yes
|
Self-dual: |
yes
|
Analytic rank: |
1
|
Selberg data: |
(4, 876096, ( :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 |
| 3 | | 1 |
| 13 | C2 | 1+4T+pT2 |
good | 5 | C22 | 1−4T2+p2T4 |
| 7 | C2×C2 | (1+pT2)(1+4T+pT2) |
| 11 | C22 | 1+8T2+p2T4 |
| 17 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 19 | C2×C2 | (1+4T+pT2)(1+8T+pT2) |
| 23 | C22 | 1−14T2+p2T4 |
| 29 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 31 | C2 | (1−4T+pT2)2 |
| 37 | C2×C2 | (1−10T+pT2)(1−8T+pT2) |
| 41 | C22 | 1−52T2+p2T4 |
| 43 | C2×C2 | (1−8T+pT2)(1−4T+pT2) |
| 47 | C22 | 1+80T2+p2T4 |
| 53 | C2 | (1−8T+pT2)(1+8T+pT2) |
| 59 | C22 | 1+8T2+p2T4 |
| 61 | C2×C2 | (1−8T+pT2)(1+pT2) |
| 67 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 71 | C22 | 1+128T2+p2T4 |
| 73 | C2×C2 | (1−10T+pT2)(1−4T+pT2) |
| 79 | C2×C2 | (1−4T+pT2)(1+12T+pT2) |
| 83 | C22 | 1+56T2+p2T4 |
| 89 | C22 | 1−60T2+p2T4 |
| 97 | C2×C2 | (1−2T+pT2)(1+12T+pT2) |
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
−8.011041589105212816329112188166, −7.58435567064855486686723796098, −6.83783779886561071593467447761, −6.67158334968123005796447784320, −6.25787428435554820454968357998, −5.95287424577545614572814364452, −5.24936749392889093682839921044, −4.63640353885541368604350705184, −4.13376904574689612252138819680, −3.98331010642415475247011050545, −2.86108699411148756398142603227, −2.68604415269591467528290082955, −2.21756293670217156757659842930, −0.933841790213200869561946347150, 0,
0.933841790213200869561946347150, 2.21756293670217156757659842930, 2.68604415269591467528290082955, 2.86108699411148756398142603227, 3.98331010642415475247011050545, 4.13376904574689612252138819680, 4.63640353885541368604350705184, 5.24936749392889093682839921044, 5.95287424577545614572814364452, 6.25787428435554820454968357998, 6.67158334968123005796447784320, 6.83783779886561071593467447761, 7.58435567064855486686723796098, 8.011041589105212816329112188166