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Properties

Label 8-768e4-1.1-c1e4-0-14
Degree 88
Conductor 347892350976347892350976
Sign 11
Analytic cond. 1414.331414.33
Root an. cond. 2.476392.47639
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  + 8·5-s + 12·13-s − 24·17-s + 32·25-s + 16·29-s + 12·37-s − 8·49-s + 16·53-s + 12·61-s + 96·65-s − 81-s − 192·85-s − 16·97-s + 32·101-s − 28·109-s + 24·113-s + 104·125-s + 127-s + 131-s + 137-s + 139-s + 128·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 3.57·5-s + 3.32·13-s − 5.82·17-s + 32/5·25-s + 2.97·29-s + 1.97·37-s − 8/7·49-s + 2.19·53-s + 1.53·61-s + 11.9·65-s − 1/9·81-s − 20.8·85-s − 1.62·97-s + 3.18·101-s − 2.68·109-s + 2.25·113-s + 9.30·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 10.6·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

Λ(s)=((232⋅34)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((232⋅34)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 232⋅342^{32} \cdot 3^{4}
Sign: 11
Analytic conductor: 1414.331414.33
Root analytic conductor: 2.476392.47639
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 232⋅34, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{32} \cdot 3^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 8.4191983248.419198324
L(12)L(\frac12) ≈\approx 8.4191983248.419198324
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)
bad2 1 1
3C22C_2^2 1+T4 1 + T^{4}
good5C22C_2^2 (1−4T+8T2−4pT3+p2T4)2 ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
7C22C_2^2 (1+4T2+p2T4)2 ( 1 + 4 T^{2} + p^{2} T^{4} )^{2}
11C23C_2^3 1−206T4+p4T8 1 - 206 T^{4} + p^{4} T^{8}
13C22C_2^2 (1−6T+18T2−6pT3+p2T4)2 ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}
17C2C_2 (1+6T+pT2)4 ( 1 + 6 T + p T^{2} )^{4}
19C22C_2^2×\timesC22C_2^2 (1−12T+72T2−12pT3+p2T4)(1+12T+72T2+12pT3+p2T4) ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )
23C22C_2^2 (1−38T2+p2T4)2 ( 1 - 38 T^{2} + p^{2} T^{4} )^{2}
29C22C_2^2 (1−8T+32T2−8pT3+p2T4)2 ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}
31C22C_2^2 (1+44T2+p2T4)2 ( 1 + 44 T^{2} + p^{2} T^{4} )^{2}
37C22C_2^2 (1−6T+18T2−6pT3+p2T4)2 ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}
41C2C_2 (1−8T+pT2)2(1+8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2}
43C23C_2^3 1−1198T4+p4T8 1 - 1198 T^{4} + p^{4} T^{8}
47C22C_2^2 (1+86T2+p2T4)2 ( 1 + 86 T^{2} + p^{2} T^{4} )^{2}
53C22C_2^2 (1−8T+32T2−8pT3+p2T4)2 ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}
59C22C_2^2 (1+p2T4)2 ( 1 + p^{2} T^{4} )^{2}
61C22C_2^2 (1−6T+18T2−6pT3+p2T4)2 ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}
67C23C_2^3 1+4946T4+p4T8 1 + 4946 T^{4} + p^{4} T^{8}
71C22C_2^2 (1−134T2+p2T4)2 ( 1 - 134 T^{2} + p^{2} T^{4} )^{2}
73C2C_2 (1−6T+pT2)2(1+6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2}
79C22C_2^2 (1+140T2+p2T4)2 ( 1 + 140 T^{2} + p^{2} T^{4} )^{2}
83C23C_2^3 1−5678T4+p4T8 1 - 5678 T^{4} + p^{4} T^{8}
89C22C_2^2 (1+18T2+p2T4)2 ( 1 + 18 T^{2} + p^{2} T^{4} )^{2}
97C2C_2 (1+4T+pT2)4 ( 1 + 4 T + p T^{2} )^{4}
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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

−7.17541090524028005340860416239, −6.90766429471259305559175216361, −6.60530406817808809405330585777, −6.59514634713179358768277490491, −6.55916352975344041183676534129, −6.31227264158126278770763844670, −6.11445119591430064242779320287, −5.84178694129719924994683965658, −5.72070719485549474035647467450, −5.41853600021421866967305718659, −5.13995183399375656247404544564, −4.64452540296242581733366135328, −4.55333170601221371658323845706, −4.28657100691365784008816074475, −4.27793153380316927668091195971, −3.77575529505352704074050354129, −3.37833124370845631094842673982, −2.86337915081147267109893599540, −2.56869848048385159799366707749, −2.56414720835923351115198045294, −1.97386060038264469952212255906, −1.90312982309361220561424011402, −1.79390694285162107155781629995, −0.948955856369256371994491761986, −0.851831776320703403346591185809, 0.851831776320703403346591185809, 0.948955856369256371994491761986, 1.79390694285162107155781629995, 1.90312982309361220561424011402, 1.97386060038264469952212255906, 2.56414720835923351115198045294, 2.56869848048385159799366707749, 2.86337915081147267109893599540, 3.37833124370845631094842673982, 3.77575529505352704074050354129, 4.27793153380316927668091195971, 4.28657100691365784008816074475, 4.55333170601221371658323845706, 4.64452540296242581733366135328, 5.13995183399375656247404544564, 5.41853600021421866967305718659, 5.72070719485549474035647467450, 5.84178694129719924994683965658, 6.11445119591430064242779320287, 6.31227264158126278770763844670, 6.55916352975344041183676534129, 6.59514634713179358768277490491, 6.60530406817808809405330585777, 6.90766429471259305559175216361, 7.17541090524028005340860416239

Graph of the ZZ-function along the critical line