Properties

Degree 8
Conductor $ 2^{20} \cdot 3^{8} \cdot 7^{4} $
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  − 2·49-s + 8·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯
L(s)  = 1  − 2·49-s + 8·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{20} \cdot 3^{8} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{2016} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 2^{20} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )$
$L(\frac{1}{2})$  $\approx$  $1.269999160$
$L(\frac12)$  $\approx$  $1.269999160$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;7\}$, \(F_p\) is a polynomial of degree 8. If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p$
bad2 \( 1 \)
3 \( 1 \)
7$C_2$ \( ( 1 + T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 + T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + T^{4} )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
17$C_2^2$ \( ( 1 + T^{4} )^{2} \)
19$C_2$ \( ( 1 + T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + T^{4} )^{2} \)
29$C_2$ \( ( 1 + T^{2} )^{4} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
37$C_2$ \( ( 1 + T^{2} )^{4} \)
41$C_2^2$ \( ( 1 + T^{4} )^{2} \)
43$C_2$ \( ( 1 + T^{2} )^{4} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
53$C_2$ \( ( 1 + T^{2} )^{4} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
67$C_2$ \( ( 1 + T^{2} )^{4} \)
71$C_2^2$ \( ( 1 + T^{4} )^{2} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
79$C_2$ \( ( 1 + T^{2} )^{4} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
89$C_2^2$ \( ( 1 + T^{4} )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−6.81602805952662671527989733065, −6.41620939846779624137200615489, −6.39954639714155647403833438354, −5.91417226871951285500437089425, −5.87805167280411652657719349527, −5.75967228228394173153998636269, −5.61589766192628420569618602160, −5.04820317817149915537790032745, −5.01696770359342235398471554338, −4.72460924295937248678957118743, −4.57565632907693409972007276983, −4.57235103219738370142888263103, −4.10731287785860528203320251107, −3.69769869507050417664612806438, −3.62851429915875349752060224539, −3.55734639438890375042461572094, −3.01658380050073067093677843413, −2.95097373307377632743712870248, −2.75275375171096306086207964270, −2.32073328036948585527311670018, −1.85049143501637111046529227519, −1.83053504815339100853283845117, −1.66575440404263102548913708498, −0.911336349285537840221428926381, −0.68315849343281942353411925020, 0.68315849343281942353411925020, 0.911336349285537840221428926381, 1.66575440404263102548913708498, 1.83053504815339100853283845117, 1.85049143501637111046529227519, 2.32073328036948585527311670018, 2.75275375171096306086207964270, 2.95097373307377632743712870248, 3.01658380050073067093677843413, 3.55734639438890375042461572094, 3.62851429915875349752060224539, 3.69769869507050417664612806438, 4.10731287785860528203320251107, 4.57235103219738370142888263103, 4.57565632907693409972007276983, 4.72460924295937248678957118743, 5.01696770359342235398471554338, 5.04820317817149915537790032745, 5.61589766192628420569618602160, 5.75967228228394173153998636269, 5.87805167280411652657719349527, 5.91417226871951285500437089425, 6.39954639714155647403833438354, 6.41620939846779624137200615489, 6.81602805952662671527989733065

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