L(s) = 1 | − 9-s + 8·13-s − 14·17-s + 4·37-s + 10·41-s + 6·49-s + 12·53-s + 20·61-s − 18·73-s − 8·81-s − 10·89-s + 4·97-s − 4·101-s + 12·109-s − 2·113-s − 8·117-s − 17·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 14·153-s + 157-s + 163-s + 167-s + ⋯ |
L(s) = 1 | − 1/3·9-s + 2.21·13-s − 3.39·17-s + 0.657·37-s + 1.56·41-s + 6/7·49-s + 1.64·53-s + 2.56·61-s − 2.10·73-s − 8/9·81-s − 1.05·89-s + 0.406·97-s − 0.398·101-s + 1.14·109-s − 0.188·113-s − 0.739·117-s − 1.54·121-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 + 1.13·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
Λ(s)=(=(640000s/2ΓC(s)2L(s)Λ(2−s)
Λ(s)=(=(640000s/2ΓC(s+1/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
640000
= 210⋅54
|
Sign: |
1
|
Analytic conductor: |
40.8069 |
Root analytic conductor: |
2.52745 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 640000, ( :1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
1.724888820 |
L(21) |
≈ |
1.724888820 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 5 | | 1 |
good | 3 | C22 | 1+T2+p2T4 |
| 7 | C22 | 1−6T2+p2T4 |
| 11 | C22 | 1+17T2+p2T4 |
| 13 | C2 | (1−4T+pT2)2 |
| 17 | C2 | (1+7T+pT2)2 |
| 19 | C22 | 1−7T2+p2T4 |
| 23 | C22 | 1+26T2+p2T4 |
| 29 | C2 | (1+pT2)2 |
| 31 | C22 | 1+42T2+p2T4 |
| 37 | C2 | (1−2T+pT2)2 |
| 41 | C2 | (1−5T+pT2)2 |
| 43 | C2 | (1+pT2)2 |
| 47 | C22 | 1+14T2+p2T4 |
| 53 | C2 | (1−6T+pT2)2 |
| 59 | C22 | 1+38T2+p2T4 |
| 61 | C2 | (1−10T+pT2)2 |
| 67 | C22 | 1+129T2+p2T4 |
| 71 | C22 | 1+62T2+p2T4 |
| 73 | C2 | (1+9T+pT2)2 |
| 79 | C22 | 1+138T2+p2T4 |
| 83 | C22 | 1+41T2+p2T4 |
| 89 | C2 | (1+5T+pT2)2 |
| 97 | C2 | (1−2T+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
−10.50605509965111304486465515300, −10.24993019668767040986526642586, −9.464476774997252509084814202742, −9.014783488310806066676906286092, −8.826822029803336769437210452676, −8.465430342484352186093050379282, −8.167905739342344370023067006366, −7.34755154133375962825895060082, −6.94760004129955496294405041358, −6.59803878243940616364742111278, −5.94426147354046117803981663122, −5.93614136368984664489504438907, −5.15500699103202102429843650560, −4.43230042203715345727770067725, −4.01699601747083075472287508810, −3.86599210376161220926146517499, −2.76239025809467826674546430899, −2.43843119291001336407519068539, −1.64894313034439815950593792217, −0.66921415445912516203127009829,
0.66921415445912516203127009829, 1.64894313034439815950593792217, 2.43843119291001336407519068539, 2.76239025809467826674546430899, 3.86599210376161220926146517499, 4.01699601747083075472287508810, 4.43230042203715345727770067725, 5.15500699103202102429843650560, 5.93614136368984664489504438907, 5.94426147354046117803981663122, 6.59803878243940616364742111278, 6.94760004129955496294405041358, 7.34755154133375962825895060082, 8.167905739342344370023067006366, 8.465430342484352186093050379282, 8.826822029803336769437210452676, 9.014783488310806066676906286092, 9.464476774997252509084814202742, 10.24993019668767040986526642586, 10.50605509965111304486465515300