L(s) = 1 | − 7-s − 9-s + 2·11-s + 6·23-s − 2·25-s − 4·29-s − 4·37-s + 4·43-s + 49-s + 8·53-s + 63-s + 16·67-s − 6·71-s − 2·77-s + 16·79-s + 81-s − 2·99-s − 6·107-s − 20·109-s − 4·113-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
L(s) = 1 | − 0.377·7-s − 1/3·9-s + 0.603·11-s + 1.25·23-s − 2/5·25-s − 0.742·29-s − 0.657·37-s + 0.609·43-s + 1/7·49-s + 1.09·53-s + 0.125·63-s + 1.95·67-s − 0.712·71-s − 0.227·77-s + 1.80·79-s + 1/9·81-s − 0.201·99-s − 0.580·107-s − 1.91·109-s − 0.376·113-s − 0.909·121-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 + ⋯ |
Λ(s)=(=(395136s/2ΓC(s)2L(s)Λ(2−s)
Λ(s)=(=(395136s/2ΓC(s+1/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
395136
= 27⋅32⋅73
|
Sign: |
1
|
Analytic conductor: |
25.1942 |
Root analytic conductor: |
2.24039 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
yes
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 395136, ( :1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
1.634075498 |
L(21) |
≈ |
1.634075498 |
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+T2 |
| 7 | C1 | 1+T |
good | 5 | C22 | 1+2T2+p2T4 |
| 11 | C2×C2 | (1−4T+pT2)(1+2T+pT2) |
| 13 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 17 | C22 | 1+26T2+p2T4 |
| 19 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 23 | C2×C2 | (1−8T+pT2)(1+2T+pT2) |
| 29 | C2×C2 | (1+pT2)(1+4T+pT2) |
| 31 | C2 | (1−10T+pT2)(1+10T+pT2) |
| 37 | C2 | (1+2T+pT2)2 |
| 41 | C22 | 1+2T2+p2T4 |
| 43 | C2×C2 | (1−4T+pT2)(1+pT2) |
| 47 | C22 | 1+26T2+p2T4 |
| 53 | C2×C2 | (1−6T+pT2)(1−2T+pT2) |
| 59 | C22 | 1+38T2+p2T4 |
| 61 | C22 | 1−42T2+p2T4 |
| 67 | C2×C2 | (1−12T+pT2)(1−4T+pT2) |
| 71 | C2×C2 | (1−6T+pT2)(1+12T+pT2) |
| 73 | C22 | 1+78T2+p2T4 |
| 79 | C2×C2 | (1−16T+pT2)(1+pT2) |
| 83 | C22 | 1−86T2+p2T4 |
| 89 | C22 | 1+42T2+p2T4 |
| 97 | C22 | 1+22T2+p2T4 |
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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
−8.699282088321589201467643517093, −8.236972050375983718950285796462, −7.77018949902582179915957339205, −7.14070251885670841829509075166, −6.87700863147116782395497369892, −6.39344426492772010571509299168, −5.80637317277794281955672583724, −5.36007472380367461897506998623, −4.92880193215004895162898338699, −4.11584250552478313262890669286, −3.76479384395257081325203691637, −3.13250433380878648175129575563, −2.49956093243554054195671052448, −1.73125840677039039003718737465, −0.72046474937322208138838996608,
0.72046474937322208138838996608, 1.73125840677039039003718737465, 2.49956093243554054195671052448, 3.13250433380878648175129575563, 3.76479384395257081325203691637, 4.11584250552478313262890669286, 4.92880193215004895162898338699, 5.36007472380367461897506998623, 5.80637317277794281955672583724, 6.39344426492772010571509299168, 6.87700863147116782395497369892, 7.14070251885670841829509075166, 7.77018949902582179915957339205, 8.236972050375983718950285796462, 8.699282088321589201467643517093