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Properties

Label 8-168e4-1.1-c1e4-0-6
Degree $8$
Conductor $796594176$
Sign $1$
Analytic cond. $3.23851$
Root an. cond. $1.15822$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·3-s − 2·4-s − 6·5-s − 2·7-s + 21·9-s + 6·11-s − 12·12-s − 36·15-s + 12·20-s − 12·21-s + 17·25-s + 54·27-s + 4·28-s + 36·29-s − 30·31-s + 36·33-s + 12·35-s − 42·36-s − 12·44-s − 126·45-s + 7·49-s + 6·53-s − 36·55-s − 18·59-s + 72·60-s − 42·63-s + 8·64-s + ⋯
L(s)  = 1  + 3.46·3-s − 4-s − 2.68·5-s − 0.755·7-s + 7·9-s + 1.80·11-s − 3.46·12-s − 9.29·15-s + 2.68·20-s − 2.61·21-s + 17/5·25-s + 10.3·27-s + 0.755·28-s + 6.68·29-s − 5.38·31-s + 6.26·33-s + 2.02·35-s − 7·36-s − 1.80·44-s − 18.7·45-s + 49-s + 0.824·53-s − 4.85·55-s − 2.34·59-s + 9.29·60-s − 5.29·63-s + 64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(3.23851\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 3^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.370212305\)
\(L(\frac12)\) \(\approx\) \(2.370212305\)
\(L(\frac{3}{2})\) 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)$
bad2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
3$C_2$ \( ( 1 - p T + p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
good5$C_2^2$$\times$$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \)
11$C_2^2$$\times$$C_2^2$ \( ( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 - 10 T^{2} + p^{2} T^{4} ) \)
13$C_2$ \( ( 1 + p T^{2} )^{4} \)
17$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
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$C_2^2$ \( ( 1 - 18 T + 137 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2$$\times$$C_2^2$ \( ( 1 + 10 T + p T^{2} )^{2}( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \)
37$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{4} \)
43$C_2$ \( ( 1 - p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$$\times$$C_2^2$ \( ( 1 - 6 T + 65 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 - 94 T^{2} + p^{2} T^{4} ) \)
59$C_2^2$$\times$$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )( 1 + 18 T + 167 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \)
61$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
71$C_2$ \( ( 1 - p T^{2} )^{4} \)
73$C_2^2$$\times$$C_2^2$ \( ( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \)
79$C_2$$\times$$C_2^2$ \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \)
83$C_2^2$$\times$$C_2^2$ \( ( 1 - 30 T + 383 T^{2} - 30 p T^{3} + p^{2} T^{4} )( 1 + 30 T + 383 T^{2} + 30 p T^{3} + p^{2} T^{4} ) \)
89$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$$\times$$C_2^2$ \( ( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \)
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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

−9.081614607006918424209978112388, −8.984240680934966648402970104489, −8.862718452280285750200296245631, −8.690658367883543879252176892806, −8.306887473294608186943884842203, −8.137653966365733706642113049803, −7.916220613973956293942346581565, −7.70343024205803142836580164284, −7.19625598872623055651045968226, −7.12005730994234531843561402131, −6.87418260018588715958806117078, −6.62971602430470037355028976819, −6.20966689603491368244029360232, −5.58607104680017025375238457088, −4.85394360141271860217538016938, −4.64103339255794857769084713322, −4.39130656411964321773884536855, −4.00069423234993990684982530594, −3.89308266053694064024751807343, −3.58602435242247003626243127541, −3.15682033607213618155360099238, −3.12843816523757430837690512104, −2.53490431193428624634432690307, −1.85612458809570943607386661198, −1.02144656163216996523133460833, 1.02144656163216996523133460833, 1.85612458809570943607386661198, 2.53490431193428624634432690307, 3.12843816523757430837690512104, 3.15682033607213618155360099238, 3.58602435242247003626243127541, 3.89308266053694064024751807343, 4.00069423234993990684982530594, 4.39130656411964321773884536855, 4.64103339255794857769084713322, 4.85394360141271860217538016938, 5.58607104680017025375238457088, 6.20966689603491368244029360232, 6.62971602430470037355028976819, 6.87418260018588715958806117078, 7.12005730994234531843561402131, 7.19625598872623055651045968226, 7.70343024205803142836580164284, 7.916220613973956293942346581565, 8.137653966365733706642113049803, 8.306887473294608186943884842203, 8.690658367883543879252176892806, 8.862718452280285750200296245631, 8.984240680934966648402970104489, 9.081614607006918424209978112388

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