L(s) = 1 | + 5-s + 4·7-s + 4·11-s + 2·13-s − 4·17-s + 8·19-s + 4·23-s − 2·29-s + 8·31-s + 4·35-s + 12·37-s − 6·41-s + 8·43-s + 4·47-s + 7·49-s − 12·53-s + 4·55-s − 4·59-s + 2·61-s + 2·65-s − 8·67-s − 12·73-s + 16·77-s − 16·83-s − 4·85-s + 12·89-s + 8·91-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.51·7-s + 1.20·11-s + 0.554·13-s − 0.970·17-s + 1.83·19-s + 0.834·23-s − 0.371·29-s + 1.43·31-s + 0.676·35-s + 1.97·37-s − 0.937·41-s + 1.21·43-s + 0.583·47-s + 49-s − 1.64·53-s + 0.539·55-s − 0.520·59-s + 0.256·61-s + 0.248·65-s − 0.977·67-s − 1.40·73-s + 1.82·77-s − 1.75·83-s − 0.433·85-s + 1.27·89-s + 0.838·91-s + ⋯ |
Λ(s)=(=(10497600s/2ΓC(s)2L(s)Λ(2−s)
Λ(s)=(=(10497600s/2ΓC(s+1/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
10497600
= 26⋅38⋅52
|
Sign: |
1
|
Analytic conductor: |
669.336 |
Root analytic conductor: |
5.08640 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 10497600, ( :1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
5.434640946 |
L(21) |
≈ |
5.434640946 |
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 |
| 5 | C2 | 1−T+T2 |
good | 7 | C2 | (1−5T+pT2)(1+T+pT2) |
| 11 | C22 | 1−4T+5T2−4pT3+p2T4 |
| 13 | C2 | (1−7T+pT2)(1+5T+pT2) |
| 17 | C2 | (1+2T+pT2)2 |
| 19 | C2 | (1−4T+pT2)2 |
| 23 | C22 | 1−4T−7T2−4pT3+p2T4 |
| 29 | C22 | 1+2T−25T2+2pT3+p2T4 |
| 31 | C22 | 1−8T+33T2−8pT3+p2T4 |
| 37 | C2 | (1−6T+pT2)2 |
| 41 | C22 | 1+6T−5T2+6pT3+p2T4 |
| 43 | C2 | (1−13T+pT2)(1+5T+pT2) |
| 47 | C22 | 1−4T−31T2−4pT3+p2T4 |
| 53 | C2 | (1+6T+pT2)2 |
| 59 | C22 | 1+4T−43T2+4pT3+p2T4 |
| 61 | C22 | 1−2T−57T2−2pT3+p2T4 |
| 67 | C22 | 1+8T−3T2+8pT3+p2T4 |
| 71 | C2 | (1+pT2)2 |
| 73 | C2 | (1+6T+pT2)2 |
| 79 | C22 | 1−pT2+p2T4 |
| 83 | C22 | 1+16T+173T2+16pT3+p2T4 |
| 89 | C2 | (1−6T+pT2)2 |
| 97 | C2 | (1−19T+pT2)(1+5T+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
−8.775738221971129928713017847475, −8.620925990908311200375000500775, −8.055497835935780895990818239749, −7.70580819100260057630002073703, −7.39564614325142566789176250416, −7.10097507023005863153467861656, −6.45354592331089491291581107900, −6.25473549548713818342062133531, −5.79283828259738763106229961254, −5.52935531655825512705437523890, −4.77255057291672011978712871596, −4.66002107074911457044767329640, −4.39401867612762282632330780686, −3.79866559309019276402115184434, −3.14499468958761546637218678166, −2.92894620361717558564372310535, −2.16028693128308461113817397367, −1.71580801263595136428293652893, −1.15445220581255516115920080329, −0.855327904736472794737847602444,
0.855327904736472794737847602444, 1.15445220581255516115920080329, 1.71580801263595136428293652893, 2.16028693128308461113817397367, 2.92894620361717558564372310535, 3.14499468958761546637218678166, 3.79866559309019276402115184434, 4.39401867612762282632330780686, 4.66002107074911457044767329640, 4.77255057291672011978712871596, 5.52935531655825512705437523890, 5.79283828259738763106229961254, 6.25473549548713818342062133531, 6.45354592331089491291581107900, 7.10097507023005863153467861656, 7.39564614325142566789176250416, 7.70580819100260057630002073703, 8.055497835935780895990818239749, 8.620925990908311200375000500775, 8.775738221971129928713017847475