Properties

Degree $8$
Conductor $2.252\times 10^{12}$
Sign $1$
Motivic weight $0$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·11-s + 16-s − 8·71-s + 81-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 4·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  − 4·11-s + 16-s − 8·71-s + 81-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 4·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{8} \cdot 7^{8}\)
Sign: $1$
Motivic weight: \(0\)
Character: induced by $\chi_{1225} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5^{8} \cdot 7^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4460828518\)
\(L(\frac12)\) \(\approx\) \(0.4460828518\)
\(L(1)\) 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)$
bad5 \( 1 \)
7 \( 1 \)
good2$C_2^3$ \( 1 - T^{4} + T^{8} \)
3$C_2^3$ \( 1 - T^{4} + T^{8} \)
11$C_2$ \( ( 1 + T + T^{2} )^{4} \)
13$C_2^2$ \( ( 1 + T^{4} )^{2} \)
17$C_2^3$ \( 1 - T^{4} + T^{8} \)
19$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
23$C_2^3$ \( 1 - T^{4} + T^{8} \)
29$C_2$ \( ( 1 + T^{2} )^{4} \)
31$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
37$C_2^3$ \( 1 - T^{4} + T^{8} \)
41$C_2$ \( ( 1 + T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + T^{4} )^{2} \)
47$C_2^3$ \( 1 - T^{4} + T^{8} \)
53$C_2^3$ \( 1 - T^{4} + T^{8} \)
59$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
61$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
67$C_2^3$ \( 1 - T^{4} + T^{8} \)
71$C_1$ \( ( 1 + T )^{8} \)
73$C_2^3$ \( 1 - T^{4} + T^{8} \)
79$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
97$C_2^2$ \( ( 1 + T^{4} )^{2} \)
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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

−7.36073443779792664314877648857, −6.93727355747591879863049198622, −6.88038970613669655230438498838, −6.52911788623658706797619758861, −6.05173452693762851578185271129, −5.98991009960892986026453138808, −5.67138986361282025414797963375, −5.62122261787881792230381473715, −5.61815753668855137836644432381, −5.06673423761891086002556058371, −4.97329414844732307864484567044, −4.73532175608862269840415202864, −4.47768030105352114777409982647, −4.36886402937230402016931794056, −3.94965068992227505423442553231, −3.49584534378517935283434112741, −3.42400830224569176083762994909, −2.97780946952674526416349711622, −2.74191640962392688143513930944, −2.66488087310503864355433301927, −2.58310175237063391028245005097, −1.80357785514661503823830905434, −1.76493036919057454849486216818, −1.29453476774021505052523383517, −0.44822589248838267746703118499, 0.44822589248838267746703118499, 1.29453476774021505052523383517, 1.76493036919057454849486216818, 1.80357785514661503823830905434, 2.58310175237063391028245005097, 2.66488087310503864355433301927, 2.74191640962392688143513930944, 2.97780946952674526416349711622, 3.42400830224569176083762994909, 3.49584534378517935283434112741, 3.94965068992227505423442553231, 4.36886402937230402016931794056, 4.47768030105352114777409982647, 4.73532175608862269840415202864, 4.97329414844732307864484567044, 5.06673423761891086002556058371, 5.61815753668855137836644432381, 5.62122261787881792230381473715, 5.67138986361282025414797963375, 5.98991009960892986026453138808, 6.05173452693762851578185271129, 6.52911788623658706797619758861, 6.88038970613669655230438498838, 6.93727355747591879863049198622, 7.36073443779792664314877648857

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