L(s) = 1 | − 2·7-s − 3·9-s − 10·17-s + 6·23-s − 6·25-s − 20·31-s − 12·41-s − 16·47-s − 11·49-s + 6·63-s + 24·71-s + 22·73-s − 32·79-s − 8·89-s − 24·97-s + 4·103-s + 16·113-s + 20·119-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 30·153-s + 157-s + ⋯ |
L(s) = 1 | − 0.755·7-s − 9-s − 2.42·17-s + 1.25·23-s − 6/5·25-s − 3.59·31-s − 1.87·41-s − 2.33·47-s − 1.57·49-s + 0.755·63-s + 2.84·71-s + 2.57·73-s − 3.60·79-s − 0.847·89-s − 2.43·97-s + 0.394·103-s + 1.50·113-s + 1.83·119-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 2.42·153-s + 0.0798·157-s + ⋯ |
Λ(s)=(=(1478656s/2ΓC(s)2L(s)Λ(2−s)
Λ(s)=(=(1478656s/2ΓC(s+1/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
1478656
= 212⋅192
|
Sign: |
1
|
Analytic conductor: |
94.2803 |
Root analytic conductor: |
3.11605 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
2
|
Selberg data: |
(4, 1478656, ( :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 |
| 19 | C2 | 1+T2 |
good | 3 | C22 | 1+pT2+p2T4 |
| 5 | C2 | (1−2T+pT2)(1+2T+pT2) |
| 7 | C2 | (1+T+pT2)2 |
| 11 | C2 | (1−pT2)2 |
| 13 | C22 | 1−T2+p2T4 |
| 17 | C2 | (1+5T+pT2)2 |
| 23 | C2 | (1−3T+pT2)2 |
| 29 | C22 | 1−9T2+p2T4 |
| 31 | C2 | (1+10T+pT2)2 |
| 37 | C2 | (1−12T+pT2)(1+12T+pT2) |
| 41 | C2 | (1+6T+pT2)2 |
| 43 | C22 | 1−70T2+p2T4 |
| 47 | C2 | (1+8T+pT2)2 |
| 53 | C22 | 1−25T2+p2T4 |
| 59 | C22 | 1−117T2+p2T4 |
| 61 | C22 | 1−118T2+p2T4 |
| 67 | C22 | 1−85T2+p2T4 |
| 71 | C2 | (1−12T+pT2)2 |
| 73 | C2 | (1−11T+pT2)2 |
| 79 | C2 | (1+16T+pT2)2 |
| 83 | C22 | 1+30T2+p2T4 |
| 89 | C2 | (1+4T+pT2)2 |
| 97 | C2 | (1+12T+pT2)2 |
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
−9.560423801687851858035516854593, −9.078914580800521084834807858492, −8.789738150830262684436493570506, −8.364989284097332836323452357892, −8.054932625377401236969486030192, −7.37759537552068899213166869092, −6.81893578599937410538296245717, −6.75490869545100678888110266108, −6.33333972766039216673932136650, −5.64627389990694439814382002943, −5.31229610751105295105052134212, −4.94020677998479479840763424810, −4.28806962137891571388993652914, −3.68374437429460524000993398476, −3.34014982079073240885196077070, −2.83619747990884525794359393850, −1.97657079152034106338366274028, −1.77849389118508217197657670653, 0, 0,
1.77849389118508217197657670653, 1.97657079152034106338366274028, 2.83619747990884525794359393850, 3.34014982079073240885196077070, 3.68374437429460524000993398476, 4.28806962137891571388993652914, 4.94020677998479479840763424810, 5.31229610751105295105052134212, 5.64627389990694439814382002943, 6.33333972766039216673932136650, 6.75490869545100678888110266108, 6.81893578599937410538296245717, 7.37759537552068899213166869092, 8.054932625377401236969486030192, 8.364989284097332836323452357892, 8.789738150830262684436493570506, 9.078914580800521084834807858492, 9.560423801687851858035516854593