Properties

Label 8-507e4-1.1-c1e4-0-9
Degree $8$
Conductor $66074188401$
Sign $1$
Analytic cond. $268.621$
Root an. cond. $2.01206$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 3·4-s + 9-s − 6·12-s + 4·16-s + 4·17-s + 12·25-s − 2·27-s + 20·29-s − 3·36-s − 24·43-s + 8·48-s + 2·49-s + 8·51-s + 24·53-s + 4·61-s − 9·64-s − 12·68-s + 24·75-s + 32·79-s − 4·81-s + 40·87-s − 36·100-s − 36·101-s − 24·107-s + 6·108-s + 12·113-s + ⋯
L(s)  = 1  + 1.15·3-s − 3/2·4-s + 1/3·9-s − 1.73·12-s + 16-s + 0.970·17-s + 12/5·25-s − 0.384·27-s + 3.71·29-s − 1/2·36-s − 3.65·43-s + 1.15·48-s + 2/7·49-s + 1.12·51-s + 3.29·53-s + 0.512·61-s − 9/8·64-s − 1.45·68-s + 2.77·75-s + 3.60·79-s − 4/9·81-s + 4.28·87-s − 3.59·100-s − 3.58·101-s − 2.32·107-s + 0.577·108-s + 1.12·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(268.621\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 13^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.588126141\)
\(L(\frac12)\) \(\approx\) \(2.588126141\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_2$ \( ( 1 - T + T^{2} )^{2} \)
13 \( 1 \)
good2$C_2^3$ \( 1 + 3 T^{2} + 5 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} \)
5$C_2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \)
7$C_2^2$$\times$$C_2^2$ \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \)
11$C_2^3$ \( 1 + 6 T^{2} - 85 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2^2$ \( ( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$$\times$$C_2^2$ \( ( 1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4} )( 1 + 12 T + 107 T^{2} + 12 p T^{3} + p^{2} T^{4} ) \)
41$C_2^3$ \( 1 + 46 T^{2} + 435 T^{4} + 46 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2^2$ \( ( 1 + 12 T + 101 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2$ \( ( 1 - p T^{2} )^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
59$C_2^3$ \( 1 - 26 T^{2} - 2805 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^3$ \( 1 + 70 T^{2} + 411 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^3$ \( 1 + 174 T^{2} + 22355 T^{4} + 174 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^3$ \( 1 + 94 T^{2} - 573 T^{4} + 94 p^{2} T^{6} + p^{4} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.967242877284151867198205796654, −7.914854382130781075196214412133, −7.45832661107519511001384848081, −7.26902923345936071442047001961, −6.76475613307462198498448976832, −6.60092293847460045573578207566, −6.59380417041481276008485035464, −6.40896089649131685023041641185, −5.89703178190644632425152677084, −5.32759007511666517854498648748, −5.17828559136724514226844144433, −5.15793714502192657356880132517, −5.03343672835818185234682035605, −4.50907840137758826199335631566, −4.30324775280008398674875661608, −3.90362739056643014903956334910, −3.84388913374310136613066332041, −3.23733454865177329590224391855, −3.20042096471468075968969185059, −2.77682253026464955026954085100, −2.57776087182548201477794171579, −2.22035890197502677009601545077, −1.32706513678421127496192652455, −1.21258031371111450142897206751, −0.58427601069559211659278518920, 0.58427601069559211659278518920, 1.21258031371111450142897206751, 1.32706513678421127496192652455, 2.22035890197502677009601545077, 2.57776087182548201477794171579, 2.77682253026464955026954085100, 3.20042096471468075968969185059, 3.23733454865177329590224391855, 3.84388913374310136613066332041, 3.90362739056643014903956334910, 4.30324775280008398674875661608, 4.50907840137758826199335631566, 5.03343672835818185234682035605, 5.15793714502192657356880132517, 5.17828559136724514226844144433, 5.32759007511666517854498648748, 5.89703178190644632425152677084, 6.40896089649131685023041641185, 6.59380417041481276008485035464, 6.60092293847460045573578207566, 6.76475613307462198498448976832, 7.26902923345936071442047001961, 7.45832661107519511001384848081, 7.914854382130781075196214412133, 7.967242877284151867198205796654

Graph of the $Z$-function along the critical line