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Properties

Label 8-5e16-1.1-c1e4-0-2
Degree 88
Conductor 152587890625152587890625
Sign 11
Analytic cond. 620.338620.338
Root an. cond. 2.233972.23397
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 4·3-s + 2·4-s − 8·6-s + 2·7-s + 5·8-s + 13·9-s + 3·11-s − 8·12-s − 9·13-s + 4·14-s + 5·16-s + 7·17-s + 26·18-s + 5·19-s − 8·21-s + 6·22-s − 9·23-s − 20·24-s − 18·26-s − 30·27-s + 4·28-s − 2·31-s − 2·32-s − 12·33-s + 14·34-s + 26·36-s + ⋯
L(s)  = 1  + 1.41·2-s − 2.30·3-s + 4-s − 3.26·6-s + 0.755·7-s + 1.76·8-s + 13/3·9-s + 0.904·11-s − 2.30·12-s − 2.49·13-s + 1.06·14-s + 5/4·16-s + 1.69·17-s + 6.12·18-s + 1.14·19-s − 1.74·21-s + 1.27·22-s − 1.87·23-s − 4.08·24-s − 3.53·26-s − 5.77·27-s + 0.755·28-s − 0.359·31-s − 0.353·32-s − 2.08·33-s + 2.40·34-s + 13/3·36-s + ⋯

Functional equation

Λ(s)=((516)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((516)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 5165^{16}
Sign: 11
Analytic conductor: 620.338620.338
Root analytic conductor: 2.233972.23397
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 516, ( :1/2,1/2,1/2,1/2), 1)(8,\ 5^{16} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 3.9095908263.909590826
L(12)L(\frac12) ≈\approx 3.9095908263.909590826
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad5 1 1
good2C22:C4C_2^2:C_4 1−pT+pT2−5T3+11T4−5pT5+p3T6−p4T7+p4T8 1 - p T + p T^{2} - 5 T^{3} + 11 T^{4} - 5 p T^{5} + p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8}
3C22:C4C_2^2:C_4 1+4T+pT2−10T3−29T4−10pT5+p3T6+4p3T7+p4T8 1 + 4 T + p T^{2} - 10 T^{3} - 29 T^{4} - 10 p T^{5} + p^{3} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}
7D4D_{4} (1−T+3T2−pT3+p2T4)2 ( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} )^{2}
11C4C_4×\timesC4C_4 (1−4T+6T2−4pT3+p2T4)(1+T−9T2+pT3+p2T4) ( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + T - 9 T^{2} + p T^{3} + p^{2} T^{4} )
13C22:C4C_2^2:C_4 1+9T+48T2+235T3+1011T4+235pT5+48p2T6+9p3T7+p4T8 1 + 9 T + 48 T^{2} + 235 T^{3} + 1011 T^{4} + 235 p T^{5} + 48 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8}
17C22:C4C_2^2:C_4 1−7T+52T2−245T3+1311T4−245pT5+52p2T6−7p3T7+p4T8 1 - 7 T + 52 T^{2} - 245 T^{3} + 1311 T^{4} - 245 p T^{5} + 52 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8}
19C22:C4C_2^2:C_4 1−5T+21T2−145T3+956T4−145pT5+21p2T6−5p3T7+p4T8 1 - 5 T + 21 T^{2} - 145 T^{3} + 956 T^{4} - 145 p T^{5} + 21 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8}
23C22:C4C_2^2:C_4 1+9T+13T2−15T3+196T4−15pT5+13p2T6+9p3T7+p4T8 1 + 9 T + 13 T^{2} - 15 T^{3} + 196 T^{4} - 15 p T^{5} + 13 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8}
29C22:C4C_2^2:C_4 1+11T2+90T3+661T4+90pT5+11p2T6+p4T8 1 + 11 T^{2} + 90 T^{3} + 661 T^{4} + 90 p T^{5} + 11 p^{2} T^{6} + p^{4} T^{8}
31C4×C2C_4\times C_2 1+2T−27T2−116T3+605T4−116pT5−27p2T6+2p3T7+p4T8 1 + 2 T - 27 T^{2} - 116 T^{3} + 605 T^{4} - 116 p T^{5} - 27 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
37C4×C2C_4\times C_2 1+3T−28T2−195T3+451T4−195pT5−28p2T6+3p3T7+p4T8 1 + 3 T - 28 T^{2} - 195 T^{3} + 451 T^{4} - 195 p T^{5} - 28 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}
41C22:C4C_2^2:C_4 1−8T+23T2−356T3+3905T4−356pT5+23p2T6−8p3T7+p4T8 1 - 8 T + 23 T^{2} - 356 T^{3} + 3905 T^{4} - 356 p T^{5} + 23 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}
43D4D_{4} (1−8T+82T2−8pT3+p2T4)2 ( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}
47C22:C4C_2^2:C_4 1+13T+67T2−115T3−3864T4−115pT5+67p2T6+13p3T7+p4T8 1 + 13 T + 67 T^{2} - 115 T^{3} - 3864 T^{4} - 115 p T^{5} + 67 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8}
53C22:C4C_2^2:C_4 1+14T+23T2−400T3−2899T4−400pT5+23p2T6+14p3T7+p4T8 1 + 14 T + 23 T^{2} - 400 T^{3} - 2899 T^{4} - 400 p T^{5} + 23 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8}
59C22:C4C_2^2:C_4 1+15T+206T2+1965T3+18601T4+1965pT5+206p2T6+15p3T7+p4T8 1 + 15 T + 206 T^{2} + 1965 T^{3} + 18601 T^{4} + 1965 p T^{5} + 206 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8}
61C22:C4C_2^2:C_4 1+2T+3T2+424T3+4265T4+424pT5+3p2T6+2p3T7+p4T8 1 + 2 T + 3 T^{2} + 424 T^{3} + 4265 T^{4} + 424 p T^{5} + 3 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
67C22:C4C_2^2:C_4 1−17T+42T2+1025T3−12559T4+1025pT5+42p2T6−17p3T7+p4T8 1 - 17 T + 42 T^{2} + 1025 T^{3} - 12559 T^{4} + 1025 p T^{5} + 42 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8}
71C22:C4C_2^2:C_4 1−13T+133T2−1541T3+17940T4−1541pT5+133p2T6−13p3T7+p4T8 1 - 13 T + 133 T^{2} - 1541 T^{3} + 17940 T^{4} - 1541 p T^{5} + 133 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8}
73C22:C4C_2^2:C_4 1−11T−27T2+515T3−364T4+515pT5−27p2T6−11p3T7+p4T8 1 - 11 T - 27 T^{2} + 515 T^{3} - 364 T^{4} + 515 p T^{5} - 27 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8}
79C4×C2C_4\times C_2 1−15T+56T2−675T3+11821T4−675pT5+56p2T6−15p3T7+p4T8 1 - 15 T + 56 T^{2} - 675 T^{3} + 11821 T^{4} - 675 p T^{5} + 56 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8}
83C22:C4C_2^2:C_4 1+24T+173T2+360T3+361T4+360pT5+173p2T6+24p3T7+p4T8 1 + 24 T + 173 T^{2} + 360 T^{3} + 361 T^{4} + 360 p T^{5} + 173 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8}
89C22:C4C_2^2:C_4 1−79T2−420T3+7501T4−420pT5−79p2T6+p4T8 1 - 79 T^{2} - 420 T^{3} + 7501 T^{4} - 420 p T^{5} - 79 p^{2} T^{6} + p^{4} T^{8}
97C22:C4C_2^2:C_4 1−12T−3T2+1220T3−13419T4+1220pT5−3p2T6−12p3T7+p4T8 1 - 12 T - 3 T^{2} + 1220 T^{3} - 13419 T^{4} + 1220 p T^{5} - 3 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}
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   L(s)=∏p ∏j=18(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−7.47836318088706972732893084734, −7.33600057268098509361781271926, −7.27065474882939287827323978196, −6.82931178072249117282980337294, −6.56294181489535797219594877233, −6.30514150552568834721157066857, −6.26635016774119225323197472347, −5.91599592953959347989224756091, −5.52405395929281706886118680218, −5.29068520051448658715331020709, −5.25351622840027320775193644875, −4.85104492359748857139906578976, −4.75503919582521384755757918915, −4.64714061143781013976389274238, −4.32841469839764762129920312591, −4.04707876825926941313672645168, −3.88777033794191258070430882843, −3.35623568282945456974879732985, −3.25482761404315604856614474424, −2.61463517132116180693321874054, −1.88265087388517799331158264316, −1.86671081774407816360712955642, −1.80619808202100508086949096656, −0.894556500888698653190925372447, −0.67382017456348229249607518921, 0.67382017456348229249607518921, 0.894556500888698653190925372447, 1.80619808202100508086949096656, 1.86671081774407816360712955642, 1.88265087388517799331158264316, 2.61463517132116180693321874054, 3.25482761404315604856614474424, 3.35623568282945456974879732985, 3.88777033794191258070430882843, 4.04707876825926941313672645168, 4.32841469839764762129920312591, 4.64714061143781013976389274238, 4.75503919582521384755757918915, 4.85104492359748857139906578976, 5.25351622840027320775193644875, 5.29068520051448658715331020709, 5.52405395929281706886118680218, 5.91599592953959347989224756091, 6.26635016774119225323197472347, 6.30514150552568834721157066857, 6.56294181489535797219594877233, 6.82931178072249117282980337294, 7.27065474882939287827323978196, 7.33600057268098509361781271926, 7.47836318088706972732893084734

Graph of the ZZ-function along the critical line