L(s) = 1 | + 2·3-s + 2·4-s + 5-s − 8·7-s + 3·9-s + 6·11-s + 4·12-s − 2·13-s + 2·15-s − 6·17-s − 7·19-s + 2·20-s − 16·21-s + 10·27-s − 16·28-s + 3·29-s − 14·31-s + 12·33-s − 8·35-s + 6·36-s + 16·37-s − 4·39-s + 6·41-s + 4·43-s + 12·44-s + 3·45-s − 6·47-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 4-s + 0.447·5-s − 3.02·7-s + 9-s + 1.80·11-s + 1.15·12-s − 0.554·13-s + 0.516·15-s − 1.45·17-s − 1.60·19-s + 0.447·20-s − 3.49·21-s + 1.92·27-s − 3.02·28-s + 0.557·29-s − 2.51·31-s + 2.08·33-s − 1.35·35-s + 36-s + 2.63·37-s − 0.640·39-s + 0.937·41-s + 0.609·43-s + 1.80·44-s + 0.447·45-s − 0.875·47-s + ⋯ |
Λ(s)=(=(9025s/2ΓC(s)2L(s)Λ(2−s)
Λ(s)=(=(9025s/2ΓC(s+1/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
9025
= 52⋅192
|
Sign: |
1
|
Analytic conductor: |
0.575441 |
Root analytic conductor: |
0.870964 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 9025, ( :1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
1.386773719 |
L(21) |
≈ |
1.386773719 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 5 | C2 | 1−T+T2 |
| 19 | C2 | 1+7T+pT2 |
good | 2 | C22 | 1−pT2+p2T4 |
| 3 | C22 | 1−2T+T2−2pT3+p2T4 |
| 7 | C2 | (1+4T+pT2)2 |
| 11 | C2 | (1−3T+pT2)2 |
| 13 | C2 | (1−5T+pT2)(1+7T+pT2) |
| 17 | C22 | 1+6T+19T2+6pT3+p2T4 |
| 23 | C22 | 1−pT2+p2T4 |
| 29 | C22 | 1−3T−20T2−3pT3+p2T4 |
| 31 | C2 | (1+7T+pT2)2 |
| 37 | C2 | (1−8T+pT2)2 |
| 41 | C22 | 1−6T−5T2−6pT3+p2T4 |
| 43 | C22 | 1−4T−27T2−4pT3+p2T4 |
| 47 | C22 | 1+6T−11T2+6pT3+p2T4 |
| 53 | C22 | 1−6T−17T2−6pT3+p2T4 |
| 59 | C22 | 1−15T+166T2−15pT3+p2T4 |
| 61 | C22 | 1+5T−36T2+5pT3+p2T4 |
| 67 | C22 | 1+2T−63T2+2pT3+p2T4 |
| 71 | C22 | 1−3T−62T2−3pT3+p2T4 |
| 73 | C22 | 1+8T−9T2+8pT3+p2T4 |
| 79 | C22 | 1+5T−54T2+5pT3+p2T4 |
| 83 | C2 | (1−12T+pT2)2 |
| 89 | C22 | 1−15T+136T2−15pT3+p2T4 |
| 97 | C22 | 1+8T−33T2+8pT3+p2T4 |
show more | | |
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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
−14.45727815431112247704379615372, −13.46187718623727369698285947222, −13.23941662520608353357695345489, −12.79406976587063842094663671018, −12.40959478732948953346559269490, −11.74685280654655406145570868760, −10.80327490361199919615475316806, −10.55286062112174331753025360208, −9.547593772168487256781364363024, −9.280394832524382960342480921885, −9.239927903917167007568240168427, −8.323530410914496101847811241859, −7.03575539981179466954492587892, −6.93957105375992591277788891737, −6.33318627269780152687854974749, −6.12009696796375254811739836851, −4.25245972936901947248198134348, −3.77337017171378786291803365910, −2.76150367300491270317309970663, −2.29937073169802967093938323200,
2.29937073169802967093938323200, 2.76150367300491270317309970663, 3.77337017171378786291803365910, 4.25245972936901947248198134348, 6.12009696796375254811739836851, 6.33318627269780152687854974749, 6.93957105375992591277788891737, 7.03575539981179466954492587892, 8.323530410914496101847811241859, 9.239927903917167007568240168427, 9.280394832524382960342480921885, 9.547593772168487256781364363024, 10.55286062112174331753025360208, 10.80327490361199919615475316806, 11.74685280654655406145570868760, 12.40959478732948953346559269490, 12.79406976587063842094663671018, 13.23941662520608353357695345489, 13.46187718623727369698285947222, 14.45727815431112247704379615372