Properties

Label 8-3800e4-1.1-c0e4-0-4
Degree $8$
Conductor $2.085\times 10^{14}$
Sign $1$
Analytic cond. $12.9348$
Root an. cond. $1.37711$
Motivic weight $0$
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  + 4-s − 9-s − 4·11-s + 2·19-s − 36-s + 2·41-s − 4·44-s − 4·49-s − 2·59-s − 64-s + 2·76-s + 81-s + 4·89-s + 4·99-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 2·164-s + 167-s + 2·169-s − 2·171-s + ⋯
L(s)  = 1  + 4-s − 9-s − 4·11-s + 2·19-s − 36-s + 2·41-s − 4·44-s − 4·49-s − 2·59-s − 64-s + 2·76-s + 81-s + 4·89-s + 4·99-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 2·164-s + 167-s + 2·169-s − 2·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{8} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(12.9348\)
Root analytic conductor: \(1.37711\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 5^{8} \cdot 19^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5240511406\)
\(L(\frac12)\) \(\approx\) \(0.5240511406\)
\(L(1)\) 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)$
bad2$C_2^2$ \( 1 - T^{2} + T^{4} \)
5 \( 1 \)
19$C_2$ \( ( 1 - T + T^{2} )^{2} \)
good3$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
7$C_2$ \( ( 1 + T^{2} )^{4} \)
11$C_2$ \( ( 1 + T + T^{2} )^{4} \)
13$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
29$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
37$C_2$ \( ( 1 + T^{2} )^{4} \)
41$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
43$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
59$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
61$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
67$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
71$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
73$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
79$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
83$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
89$C_2$ \( ( 1 - T + T^{2} )^{4} \)
97$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
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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

−6.17239023837203753593964451194, −6.00875899359188988154341987657, −5.79830443028639570780183423752, −5.67209296982242134807563279161, −5.26571185475875722080018992640, −5.09369246435461083925128998982, −5.08583679416755380118742700208, −4.96694254318839599471162262452, −4.90998552981309850532456330906, −4.41663682811393252382404712365, −4.14042401226337320820489472849, −4.12743118762662828247110578370, −3.46929155713256913925837996333, −3.42188949055513843544158764582, −3.01080050816064212321032002398, −2.97607261905949125944161295219, −2.97274488362386089879285307783, −2.82838289726690583673225602858, −2.26224199376516917850708167908, −2.21192457579688578550143415161, −1.92268295060124838718515251018, −1.83794377455359326389843473585, −1.13594859349304017073477690312, −0.979565346964776444200058671677, −0.26214172676901085807433587199, 0.26214172676901085807433587199, 0.979565346964776444200058671677, 1.13594859349304017073477690312, 1.83794377455359326389843473585, 1.92268295060124838718515251018, 2.21192457579688578550143415161, 2.26224199376516917850708167908, 2.82838289726690583673225602858, 2.97274488362386089879285307783, 2.97607261905949125944161295219, 3.01080050816064212321032002398, 3.42188949055513843544158764582, 3.46929155713256913925837996333, 4.12743118762662828247110578370, 4.14042401226337320820489472849, 4.41663682811393252382404712365, 4.90998552981309850532456330906, 4.96694254318839599471162262452, 5.08583679416755380118742700208, 5.09369246435461083925128998982, 5.26571185475875722080018992640, 5.67209296982242134807563279161, 5.79830443028639570780183423752, 6.00875899359188988154341987657, 6.17239023837203753593964451194

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