L(s) = 1 | − 6·3-s − 6·5-s + 15·9-s − 4·11-s + 36·15-s + 4·17-s − 2·19-s − 18·23-s + 12·25-s − 18·27-s − 6·29-s + 12·31-s + 24·33-s − 12·37-s + 6·41-s − 90·45-s − 30·47-s + 4·49-s − 24·51-s + 12·53-s + 24·55-s + 12·57-s − 18·59-s + 18·61-s − 6·67-s + 108·69-s − 12·71-s + ⋯ |
L(s) = 1 | − 3.46·3-s − 2.68·5-s + 5·9-s − 1.20·11-s + 9.29·15-s + 0.970·17-s − 0.458·19-s − 3.75·23-s + 12/5·25-s − 3.46·27-s − 1.11·29-s + 2.15·31-s + 4.17·33-s − 1.97·37-s + 0.937·41-s − 13.4·45-s − 4.37·47-s + 4/7·49-s − 3.36·51-s + 1.64·53-s + 3.23·55-s + 1.58·57-s − 2.34·59-s + 2.30·61-s − 0.733·67-s + 13.0·69-s − 1.42·71-s + ⋯ |
Λ(s)=(=((224⋅194)s/2ΓC(s)4L(s)Λ(2−s)
Λ(s)=(=((224⋅194)s/2ΓC(s+1/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
224⋅194
|
Sign: |
1
|
Analytic conductor: |
8888.79 |
Root analytic conductor: |
3.11605 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
4
|
Selberg data: |
(8, 224⋅194, ( :1/2,1/2,1/2,1/2), 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+T+pT2)2 |
good | 3 | C2 | (1+pT2)2(1+pT+pT2)2 |
| 5 | D4×C2 | 1+6T+24T2+72T3+179T4+72pT5+24p2T6+6p3T7+p4T8 |
| 7 | D4×C2 | 1−4T2−6T4−4p2T6+p4T8 |
| 11 | D4 | (1+2T+pT2+2pT3+p2T4)2 |
| 13 | C22 | (1−pT2+p2T4)2 |
| 17 | D4×C2 | 1−4T−10T2+32T3+115T4+32pT5−10p2T6−4p3T7+p4T8 |
| 23 | D4×C2 | 1+18T+180T2+1296T3+7139T4+1296pT5+180p2T6+18p3T7+p4T8 |
| 29 | D4×C2 | 1+6T−4T2−108T3−285T4−108pT5−4p2T6+6p3T7+p4T8 |
| 31 | D4 | (1−6T+44T2−6pT3+p2T4)2 |
| 37 | D4 | (1+6T+80T2+6pT3+p2T4)2 |
| 41 | D4×C2 | 1−6T+61T2−294T3+1212T4−294pT5+61p2T6−6p3T7+p4T8 |
| 43 | C23 | 1−74T2+3627T4−74p2T6+p4T8 |
| 47 | D4×C2 | 1+30T+468T2+5040T3+40115T4+5040pT5+468p2T6+30p3T7+p4T8 |
| 53 | D4×C2 | 1−12T+14T2−288T3+6459T4−288pT5+14p2T6−12p3T7+p4T8 |
| 59 | D4×C2 | 1+18T+217T2+1962T3+14772T4+1962pT5+217p2T6+18p3T7+p4T8 |
| 61 | D4×C2 | 1−18T+208T2−1800T3+12867T4−1800pT5+208p2T6−18p3T7+p4T8 |
| 67 | D4×C2 | 1+6T+113T2+606T3+6516T4+606pT5+113p2T6+6p3T7+p4T8 |
| 71 | D4×C2 | 1+12T−22T2+288T3+12291T4+288pT5−22p2T6+12p3T7+p4T8 |
| 73 | C22 | (1−5T−48T2−5pT3+p2T4)2 |
| 79 | D4×C2 | 1+12T+58T2−864T3−10221T4−864pT5+58p2T6+12p3T7+p4T8 |
| 83 | D4 | (1+14T+107T2+14pT3+p2T4)2 |
| 89 | D4×C2 | 1+24T+382T2+4560T3+45267T4+4560pT5+382p2T6+24p3T7+p4T8 |
| 97 | D4×C2 | 1+6T+173T2+966T3+17676T4+966pT5+173p2T6+6p3T7+p4T8 |
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L(s)=p∏ j=1∏8(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−7.47764733621339253762428209427, −7.09846534150064078162454146164, −6.79268612336515928206921454207, −6.77523702346880101930218217442, −6.36297077635708366275617402124, −6.19103081538373663617802062372, −6.03074170972791034312891217699, −5.88371764144996900914480826913, −5.53950627704779727473066817125, −5.47472899284056516914550446967, −5.26383131420191214725056087745, −4.99830295761809621610591852870, −4.87985188246000690233134908678, −4.31253683958934029031937233432, −4.22563856521221089375876907575, −4.16279995631514366787115659817, −3.95040392874279238336211538598, −3.49486431902112893523638122725, −3.43498210187063801059926520658, −2.85137791666403056010258401361, −2.76398728565106356709405996052, −2.25620591739892058766358954931, −1.70730626978912369534272516663, −1.49939994273791059976384768571, −0.922295206936550637962057402849, 0, 0, 0, 0,
0.922295206936550637962057402849, 1.49939994273791059976384768571, 1.70730626978912369534272516663, 2.25620591739892058766358954931, 2.76398728565106356709405996052, 2.85137791666403056010258401361, 3.43498210187063801059926520658, 3.49486431902112893523638122725, 3.95040392874279238336211538598, 4.16279995631514366787115659817, 4.22563856521221089375876907575, 4.31253683958934029031937233432, 4.87985188246000690233134908678, 4.99830295761809621610591852870, 5.26383131420191214725056087745, 5.47472899284056516914550446967, 5.53950627704779727473066817125, 5.88371764144996900914480826913, 6.03074170972791034312891217699, 6.19103081538373663617802062372, 6.36297077635708366275617402124, 6.77523702346880101930218217442, 6.79268612336515928206921454207, 7.09846534150064078162454146164, 7.47764733621339253762428209427