L(s) = 1 | + 6·7-s + 6·19-s + 2·25-s + 6·31-s − 4·37-s + 8·43-s + 14·49-s − 20·61-s + 2·67-s + 4·79-s + 8·103-s − 8·109-s − 10·121-s + 127-s + 131-s + 36·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 12·175-s + 179-s + ⋯ |
L(s) = 1 | + 2.26·7-s + 1.37·19-s + 2/5·25-s + 1.07·31-s − 0.657·37-s + 1.21·43-s + 2·49-s − 2.56·61-s + 0.244·67-s + 0.450·79-s + 0.788·103-s − 0.766·109-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 3.12·133-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 + 1/13·169-s + 0.0760·173-s + 0.907·175-s + 0.0747·179-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: |
0
|
Selberg data: |
(4, 876096, ( :1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
3.120265992 |
L(21) |
≈ |
3.120265992 |
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 | C1×C1 | (1−T)(1+T) |
good | 5 | C22 | 1−2T2+p2T4 |
| 7 | C2×C2 | (1−4T+pT2)(1−2T+pT2) |
| 11 | C22 | 1+10T2+p2T4 |
| 17 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 19 | C2×C2 | (1−4T+pT2)(1−2T+pT2) |
| 23 | C22 | 1+26T2+p2T4 |
| 29 | C22 | 1−38T2+p2T4 |
| 31 | C2×C2 | (1−10T+pT2)(1+4T+pT2) |
| 37 | C2×C2 | (1−4T+pT2)(1+8T+pT2) |
| 41 | C22 | 1+70T2+p2T4 |
| 43 | C2 | (1−4T+pT2)2 |
| 47 | C22 | 1+26T2+p2T4 |
| 53 | C22 | 1−70T2+p2T4 |
| 59 | C2 | (1−12T+pT2)(1+12T+pT2) |
| 61 | C2 | (1+10T+pT2)2 |
| 67 | C2×C2 | (1−2T+pT2)(1+pT2) |
| 71 | C22 | 1−66T2+p2T4 |
| 73 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 79 | C2×C2 | (1−8T+pT2)(1+4T+pT2) |
| 83 | C22 | 1+118T2+p2T4 |
| 89 | C2 | (1−18T+pT2)(1+18T+pT2) |
| 97 | C2 | (1−16T+pT2)(1+16T+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
−8.075257102118721441562989676744, −7.86756139395375305198988393851, −7.41778155909519335240701448999, −7.04107829495058172241014800754, −6.40290829742781820907724539550, −5.84122750036013329338006700633, −5.38931768486352998317478144494, −4.97504874090531212686413887179, −4.55334630045497318040636110189, −4.25197876941557033379623437614, −3.37613329269396856069104097184, −2.89486831361438852855982285933, −2.10529677189616315902088168255, −1.52669754955306856650657171662, −0.934741245515398847185651189012,
0.934741245515398847185651189012, 1.52669754955306856650657171662, 2.10529677189616315902088168255, 2.89486831361438852855982285933, 3.37613329269396856069104097184, 4.25197876941557033379623437614, 4.55334630045497318040636110189, 4.97504874090531212686413887179, 5.38931768486352998317478144494, 5.84122750036013329338006700633, 6.40290829742781820907724539550, 7.04107829495058172241014800754, 7.41778155909519335240701448999, 7.86756139395375305198988393851, 8.075257102118721441562989676744