Properties

Label 8-819e4-1.1-c0e4-0-5
Degree $8$
Conductor $449920319121$
Sign $1$
Analytic cond. $0.0279102$
Root an. cond. $0.639323$
Motivic weight $0$
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  + 2·7-s + 16-s + 2·19-s − 2·31-s − 2·37-s + 6·43-s + 49-s − 4·67-s − 4·73-s − 2·97-s − 6·103-s + 2·109-s + 2·112-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 2·7-s + 16-s + 2·19-s − 2·31-s − 2·37-s + 6·43-s + 49-s − 4·67-s − 4·73-s − 2·97-s − 6·103-s + 2·109-s + 2·112-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 7^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(0.0279102\)
Root analytic conductor: \(0.639323\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 7^{4} \cdot 13^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.131654088\)
\(L(\frac12)\) \(\approx\) \(1.131654088\)
\(L(1)\) 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)$
bad3 \( 1 \)
7$C_2$ \( ( 1 - T + T^{2} )^{2} \)
13$C_2^2$ \( 1 - T^{2} + T^{4} \)
good2$C_2^3$ \( 1 - T^{4} + T^{8} \)
5$C_2^3$ \( 1 - T^{4} + T^{8} \)
11$C_2^2$ \( ( 1 + T^{4} )^{2} \)
17$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
19$C_2$$\times$$C_2^2$ \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
23$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
29$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
31$C_2$$\times$$C_2^2$ \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
37$C_2$$\times$$C_2$ \( ( 1 + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
41$C_2^3$ \( 1 - T^{4} + T^{8} \)
43$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 - T + T^{2} )^{2} \)
47$C_2^3$ \( 1 - T^{4} + T^{8} \)
53$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
59$C_2^3$ \( 1 - T^{4} + T^{8} \)
61$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
67$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
71$C_2^3$ \( 1 - T^{4} + T^{8} \)
73$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 - T^{2} + T^{4} ) \)
79$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_2^3$ \( 1 - T^{4} + T^{8} \)
97$C_2$$\times$$C_2$ \( ( 1 + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
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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

−7.57247009896421151528779520251, −7.43447111586758402498875781759, −7.25704756397129989719478031650, −7.00413122506102861104146279792, −6.95155218175250706796369691965, −6.12184918493857249235578019908, −6.08236340971301555016360185405, −5.88050261215556366916036047167, −5.85227941283572077116123002544, −5.36327450569760800441968400123, −5.14099150499034411808459670238, −5.04829324274203416652598710843, −5.02020567025159185866521503885, −4.28500326620099445323270361678, −4.11377982116627186934721578177, −4.04986296281894392663774182877, −3.93846544203856168203041988128, −3.31052947040954895929487667957, −2.94263895845517460930360134371, −2.80366753766539992535523215536, −2.62153557837044334587783572346, −2.02337167280871345113185974527, −1.45823378368166458461027255327, −1.39005318553599220502305882319, −1.27520525454857803587190793091, 1.27520525454857803587190793091, 1.39005318553599220502305882319, 1.45823378368166458461027255327, 2.02337167280871345113185974527, 2.62153557837044334587783572346, 2.80366753766539992535523215536, 2.94263895845517460930360134371, 3.31052947040954895929487667957, 3.93846544203856168203041988128, 4.04986296281894392663774182877, 4.11377982116627186934721578177, 4.28500326620099445323270361678, 5.02020567025159185866521503885, 5.04829324274203416652598710843, 5.14099150499034411808459670238, 5.36327450569760800441968400123, 5.85227941283572077116123002544, 5.88050261215556366916036047167, 6.08236340971301555016360185405, 6.12184918493857249235578019908, 6.95155218175250706796369691965, 7.00413122506102861104146279792, 7.25704756397129989719478031650, 7.43447111586758402498875781759, 7.57247009896421151528779520251

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