L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s − 7-s + 2·8-s + 9-s − 2·12-s − 14-s + 2·16-s + 17-s + 18-s + 2·21-s − 4·24-s − 25-s + 2·27-s − 28-s + 2·32-s + 34-s + 36-s + 37-s + 2·42-s − 47-s − 4·48-s − 50-s − 2·51-s + 53-s + ⋯ |
L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s − 7-s + 2·8-s + 9-s − 2·12-s − 14-s + 2·16-s + 17-s + 18-s + 2·21-s − 4·24-s − 25-s + 2·27-s − 28-s + 2·32-s + 34-s + 36-s + 37-s + 2·42-s − 47-s − 4·48-s − 50-s − 2·51-s + 53-s + ⋯ |
Λ(s)=(=(108241s/2ΓC(s)2L(s)Λ(1−s)
Λ(s)=(=(108241s/2ΓC(s)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
108241
= 72⋅472
|
Sign: |
1
|
Analytic conductor: |
0.0269591 |
Root analytic conductor: |
0.405206 |
Motivic weight: |
0 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 108241, ( :0,0), 1)
|
Particular Values
L(21) |
≈ |
0.5982080586 |
L(21) |
≈ |
0.5982080586 |
L(1) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 7 | C2 | 1+T+T2 |
| 47 | C2 | 1+T+T2 |
good | 2 | C1×C2 | (1−T)2(1+T+T2) |
| 3 | C2 | (1+T+T2)2 |
| 5 | C2 | (1−T+T2)(1+T+T2) |
| 11 | C2 | (1−T+T2)(1+T+T2) |
| 13 | C1×C1 | (1−T)2(1+T)2 |
| 17 | C1×C2 | (1−T)2(1+T+T2) |
| 19 | C2 | (1−T+T2)(1+T+T2) |
| 23 | C2 | (1−T+T2)(1+T+T2) |
| 29 | C1×C1 | (1−T)2(1+T)2 |
| 31 | C2 | (1−T+T2)(1+T+T2) |
| 37 | C1×C2 | (1−T)2(1+T+T2) |
| 41 | C1×C1 | (1−T)2(1+T)2 |
| 43 | C1×C1 | (1−T)2(1+T)2 |
| 53 | C1×C2 | (1−T)2(1+T+T2) |
| 59 | C1×C2 | (1−T)2(1+T+T2) |
| 61 | C1×C2 | (1−T)2(1+T+T2) |
| 67 | C2 | (1−T+T2)(1+T+T2) |
| 71 | C2 | (1+T+T2)2 |
| 73 | C2 | (1−T+T2)(1+T+T2) |
| 79 | C2 | (1+T+T2)2 |
| 83 | C2 | (1+T+T2)2 |
| 89 | C2 | (1+T+T2)2 |
| 97 | C2 | (1+T+T2)2 |
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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
−11.83988006959751153266928827298, −11.61522390432652797060404954170, −11.34400716457985881643436985485, −10.90672704503313158336709174108, −10.16755714907701894495099559863, −10.16158649940227520378359334706, −9.688275521645396995568294994764, −8.624463388835655697389509257478, −8.169032658583608566069655164604, −7.46253857929727232494147872029, −6.95398221082477526294204008166, −6.62079931918439691961234634925, −5.92795201129495718593740288338, −5.63318403982791156057331750624, −5.40786190130025409057174137241, −4.54907603163398575805880359415, −4.19381485711651291004855249184, −3.29721858350856258410418911185, −2.63006638421219917034747835911, −1.31151289050695141982660107552,
1.31151289050695141982660107552, 2.63006638421219917034747835911, 3.29721858350856258410418911185, 4.19381485711651291004855249184, 4.54907603163398575805880359415, 5.40786190130025409057174137241, 5.63318403982791156057331750624, 5.92795201129495718593740288338, 6.62079931918439691961234634925, 6.95398221082477526294204008166, 7.46253857929727232494147872029, 8.169032658583608566069655164604, 8.624463388835655697389509257478, 9.688275521645396995568294994764, 10.16158649940227520378359334706, 10.16755714907701894495099559863, 10.90672704503313158336709174108, 11.34400716457985881643436985485, 11.61522390432652797060404954170, 11.83988006959751153266928827298