L(s) = 1 | − 3-s − 6·5-s + 6·11-s + 6·15-s + 5·19-s − 12·23-s + 19·25-s + 27-s − 5·31-s − 6·33-s + 11·37-s − 6·47-s − 12·53-s − 36·55-s − 5·57-s + 12·59-s + 24·61-s − 15·67-s + 12·69-s + 9·73-s − 19·75-s − 21·79-s − 81-s − 36·83-s − 12·89-s + 5·93-s − 30·95-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 2.68·5-s + 1.80·11-s + 1.54·15-s + 1.14·19-s − 2.50·23-s + 19/5·25-s + 0.192·27-s − 0.898·31-s − 1.04·33-s + 1.80·37-s − 0.875·47-s − 1.64·53-s − 4.85·55-s − 0.662·57-s + 1.56·59-s + 3.07·61-s − 1.83·67-s + 1.44·69-s + 1.05·73-s − 2.19·75-s − 2.36·79-s − 1/9·81-s − 3.95·83-s − 1.27·89-s + 0.518·93-s − 3.07·95-s + ⋯ |
Λ(s)=(=(5531904s/2ΓC(s)2L(s)Λ(2−s)
Λ(s)=(=(5531904s/2ΓC(s+1/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
5531904
= 28⋅32⋅74
|
Sign: |
1
|
Analytic conductor: |
352.718 |
Root analytic conductor: |
4.33368 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 5531904, ( :1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
0.2997262932 |
L(21) |
≈ |
0.2997262932 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 3 | C2 | 1+T+T2 |
| 7 | | 1 |
good | 5 | C22 | 1+6T+17T2+6pT3+p2T4 |
| 11 | C22 | 1−6T+23T2−6pT3+p2T4 |
| 13 | C2 | (1−7T+pT2)(1+7T+pT2) |
| 17 | C22 | 1+pT2+p2T4 |
| 19 | C22 | 1−5T+6T2−5pT3+p2T4 |
| 23 | C22 | 1+12T+71T2+12pT3+p2T4 |
| 29 | C2 | (1+pT2)2 |
| 31 | C22 | 1+5T−6T2+5pT3+p2T4 |
| 37 | C2 | (1−10T+pT2)(1−T+pT2) |
| 41 | C22 | 1−70T2+p2T4 |
| 43 | C22 | 1−11T2+p2T4 |
| 47 | C22 | 1+6T−11T2+6pT3+p2T4 |
| 53 | C22 | 1+12T+91T2+12pT3+p2T4 |
| 59 | C22 | 1−12T+85T2−12pT3+p2T4 |
| 61 | C22 | 1−24T+253T2−24pT3+p2T4 |
| 67 | C22 | 1+15T+142T2+15pT3+p2T4 |
| 71 | C22 | 1−130T2+p2T4 |
| 73 | C22 | 1−9T+100T2−9pT3+p2T4 |
| 79 | C2 | (1+4T+pT2)(1+17T+pT2) |
| 83 | C2 | (1+18T+pT2)2 |
| 89 | C22 | 1+12T+137T2+12pT3+p2T4 |
| 97 | C22 | 1−146T2+p2T4 |
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
−9.671698580830243451694689247146, −8.528595555660986011144992333201, −8.449942892896759581205198928803, −7.88629172481279021040580104344, −7.78290243696478170870578388742, −7.17041631603365486434514503538, −7.06711350784399495827442739540, −6.48775595233884848843344908526, −6.20002311406581522914434435362, −5.47641920194200923920322113295, −5.42210625503084125942408110976, −4.36821327415526520552638126455, −4.35880064904406542696236399975, −3.84715448270943117777029326617, −3.84558346434773473541643891978, −3.12720229769520223265923479699, −2.63638186442779382525477860321, −1.59631233487813131159050231603, −1.12382345083544906122864893634, −0.22836289856137411672963545183,
0.22836289856137411672963545183, 1.12382345083544906122864893634, 1.59631233487813131159050231603, 2.63638186442779382525477860321, 3.12720229769520223265923479699, 3.84558346434773473541643891978, 3.84715448270943117777029326617, 4.35880064904406542696236399975, 4.36821327415526520552638126455, 5.42210625503084125942408110976, 5.47641920194200923920322113295, 6.20002311406581522914434435362, 6.48775595233884848843344908526, 7.06711350784399495827442739540, 7.17041631603365486434514503538, 7.78290243696478170870578388742, 7.88629172481279021040580104344, 8.449942892896759581205198928803, 8.528595555660986011144992333201, 9.671698580830243451694689247146