Properties

Label 8-39e8-1.1-c1e4-0-7
Degree $8$
Conductor $5.352\times 10^{12}$
Sign $1$
Analytic cond. $21758.3$
Root an. cond. $3.48500$
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  + 12·7-s − 16-s + 12·19-s − 12·31-s + 24·37-s + 72·49-s + 48·61-s + 12·67-s − 24·73-s − 24·97-s + 48·109-s − 12·112-s + 127-s + 131-s + 144·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 4.53·7-s − 1/4·16-s + 2.75·19-s − 2.15·31-s + 3.94·37-s + 72/7·49-s + 6.14·61-s + 1.46·67-s − 2.80·73-s − 2.43·97-s + 4.59·109-s − 1.13·112-s + 0.0887·127-s + 0.0873·131-s + 12.4·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(21758.3\)
Root analytic conductor: \(3.48500\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 13^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(12.75633717\)
\(L(\frac12)\) \(\approx\) \(12.75633717\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 \)
good2$C_2^3$ \( 1 + T^{4} + p^{4} T^{8} \)
5$C_2^2$$\times$$C_2^2$ \( ( 1 - 8 T^{2} + p^{2} T^{4} )( 1 + 8 T^{2} + p^{2} T^{4} ) \)
7$C_2^2$ \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2^3$ \( 1 + 82 T^{4} + p^{4} T^{8} \)
17$C_2^2$ \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2$$\times$$C_2^2$ \( ( 1 - 80 T^{2} + p^{2} T^{4} )( 1 + 80 T^{2} + p^{2} T^{4} ) \)
43$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 - 4382 T^{4} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 - 88 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^3$ \( 1 + 3442 T^{4} + p^{4} T^{8} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^3$ \( 1 - 3998 T^{4} + p^{4} T^{8} \)
73$C_2^2$ \( ( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{4} \)
83$C_2^3$ \( 1 + 8722 T^{4} + p^{4} T^{8} \)
89$C_2^3$ \( 1 + 14434 T^{4} + p^{4} T^{8} \)
97$C_2^2$ \( ( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−6.90344981795372359651462914404, −6.83475531564760493087372224980, −6.10612033833950880736900884100, −5.83141443898892663066542444221, −5.81785432624108841862543602250, −5.61180005408027724993625104784, −5.48645724344473331652853645679, −5.02540220146857899671085816069, −4.99085097428403406041131938901, −4.91891729180304149402264925014, −4.57721503621298028877561156672, −4.42428425900181351577204363506, −4.20386145538209395991200717988, −3.72403165280893903251651757098, −3.64656031164873071288596691507, −3.59733578501690052654801233112, −2.80869061157417120983899056439, −2.71686218237178907661147191220, −2.36109495861658678877652056500, −2.05332542253546676110412050236, −2.00248667890856410639425456747, −1.40282198427949415645375996686, −1.31083744063608329607057411212, −0.926474977568142480656676537087, −0.76283782296202227906796145585, 0.76283782296202227906796145585, 0.926474977568142480656676537087, 1.31083744063608329607057411212, 1.40282198427949415645375996686, 2.00248667890856410639425456747, 2.05332542253546676110412050236, 2.36109495861658678877652056500, 2.71686218237178907661147191220, 2.80869061157417120983899056439, 3.59733578501690052654801233112, 3.64656031164873071288596691507, 3.72403165280893903251651757098, 4.20386145538209395991200717988, 4.42428425900181351577204363506, 4.57721503621298028877561156672, 4.91891729180304149402264925014, 4.99085097428403406041131938901, 5.02540220146857899671085816069, 5.48645724344473331652853645679, 5.61180005408027724993625104784, 5.81785432624108841862543602250, 5.83141443898892663066542444221, 6.10612033833950880736900884100, 6.83475531564760493087372224980, 6.90344981795372359651462914404

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