Properties

Label 8-52e4-1.1-c1e4-0-0
Degree $8$
Conductor $7311616$
Sign $1$
Analytic cond. $0.0297249$
Root an. cond. $0.644377$
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  − 2·2-s + 2·4-s + 6·5-s − 4·8-s − 6·9-s − 12·10-s + 4·13-s + 8·16-s − 24·17-s + 12·18-s + 12·20-s + 18·25-s − 8·26-s + 4·29-s − 8·32-s + 48·34-s − 12·36-s + 26·37-s − 24·40-s − 28·41-s − 36·45-s − 36·50-s + 8·52-s + 28·53-s − 8·58-s + 10·61-s + 8·64-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 2.68·5-s − 1.41·8-s − 2·9-s − 3.79·10-s + 1.10·13-s + 2·16-s − 5.82·17-s + 2.82·18-s + 2.68·20-s + 18/5·25-s − 1.56·26-s + 0.742·29-s − 1.41·32-s + 8.23·34-s − 2·36-s + 4.27·37-s − 3.79·40-s − 4.37·41-s − 5.36·45-s − 5.09·50-s + 1.10·52-s + 3.84·53-s − 1.05·58-s + 1.28·61-s + 64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7311616 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7311616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(7311616\)    =    \(2^{8} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(0.0297249\)
Root analytic conductor: \(0.644377\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 7311616,\ (\ :1/2, 1/2, 1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3102919377\)
\(L(\frac12)\) \(\approx\) \(0.3102919377\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
good3$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
5$C_2^2$$\times$$C_2^2$ \( ( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} ) \)
7$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
11$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
17$C_2$$\times$$C_2^2$ \( ( 1 + 8 T + p T^{2} )^{2}( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \)
19$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
23$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
29$C_2$$\times$$C_2^2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \)
31$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
37$C_2$$\times$$C_2^2$ \( ( 1 - 12 T + p T^{2} )^{2}( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} ) \)
41$C_2$$\times$$C_2^2$ \( ( 1 + 10 T + p T^{2} )^{2}( 1 + 8 T + 23 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \)
43$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 14 T + 143 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
61$C_2$$\times$$C_2^2$ \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \)
67$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
71$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
73$C_2^2$$\times$$C_2^2$ \( ( 1 - 16 T + 183 T^{2} - 16 p T^{3} + p^{2} T^{4} )( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} ) \)
79$C_2$ \( ( 1 - p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
89$C_2^2$$\times$$C_2^2$ \( ( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} )( 1 + 16 T + 167 T^{2} + 16 p T^{3} + p^{2} T^{4} ) \)
97$C_2^2$$\times$$C_2^2$ \( ( 1 - 8 T - 33 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \)
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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

−11.54926806610758296882055014005, −11.12057149536826258913955963587, −10.91328134438960859545061080082, −10.53412890432023641970963521843, −10.53082283933156996669328733270, −9.769975835584312212024210673018, −9.571858930860050498131620442129, −9.277298427706364785580054008499, −9.261965981035636758590460985983, −8.565546882230064109159812458026, −8.486054835856890677227479457762, −8.430056323170363233677259055606, −8.297346407402089548407318600183, −6.83866115633538702196975753435, −6.77367751703942809267965650864, −6.75443969258722335577912003593, −6.15454646270486005328547451229, −5.93276301048617207890464001909, −5.66641055851929360058606293608, −5.09050989394721151313931821163, −4.53358389757030762429745569301, −3.87521082344718984008305960188, −2.62947401537451165449889642003, −2.52657693685617399564981088391, −2.11364378716024042732234045340, 2.11364378716024042732234045340, 2.52657693685617399564981088391, 2.62947401537451165449889642003, 3.87521082344718984008305960188, 4.53358389757030762429745569301, 5.09050989394721151313931821163, 5.66641055851929360058606293608, 5.93276301048617207890464001909, 6.15454646270486005328547451229, 6.75443969258722335577912003593, 6.77367751703942809267965650864, 6.83866115633538702196975753435, 8.297346407402089548407318600183, 8.430056323170363233677259055606, 8.486054835856890677227479457762, 8.565546882230064109159812458026, 9.261965981035636758590460985983, 9.277298427706364785580054008499, 9.571858930860050498131620442129, 9.769975835584312212024210673018, 10.53082283933156996669328733270, 10.53412890432023641970963521843, 10.91328134438960859545061080082, 11.12057149536826258913955963587, 11.54926806610758296882055014005

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