Properties

Degree 8
Conductor $ 5^{4} \cdot 13^{4} \cdot 31^{4} $
Sign $1$
Motivic weight 0
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s + 10·16-s − 2·25-s − 4·49-s − 20·64-s − 2·81-s + 8·100-s − 4·121-s + 127-s + 131-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 + 16·196-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  − 4·4-s + 10·16-s − 2·25-s − 4·49-s − 20·64-s − 2·81-s + 8·100-s − 4·121-s + 127-s + 131-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 + 16·196-s + 197-s + 199-s + 211-s + 223-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(5^{4} \cdot 13^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(5^{4} \cdot 13^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(5^{4} \cdot 13^{4} \cdot 31^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{2015} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 5^{4} \cdot 13^{4} \cdot 31^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )$
$L(\frac{1}{2})$  $\approx$  $0.002897918371$
$L(\frac12)$  $\approx$  $0.002897918371$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;13,\;31\}$, \(F_p\) is a polynomial of degree 8. If $p \in \{5,\;13,\;31\}$, then $F_p$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p$
bad5$C_2$ \( ( 1 + T^{2} )^{2} \)
13$C_2^2$ \( 1 + T^{4} \)
31$C_2$ \( ( 1 + T^{2} )^{2} \)
good2$C_2$ \( ( 1 + T^{2} )^{4} \)
3$C_2^2$ \( ( 1 + T^{4} )^{2} \)
7$C_2$ \( ( 1 + T^{2} )^{4} \)
11$C_2$ \( ( 1 + T^{2} )^{4} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
37$C_2^2$ \( ( 1 + T^{4} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
43$C_2^2$ \( ( 1 + T^{4} )^{2} \)
47$C_2$ \( ( 1 + T^{2} )^{4} \)
53$C_2^2$ \( ( 1 + T^{4} )^{2} \)
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$C_2$ \( ( 1 + T^{2} )^{4} \)
73$C_2^2$ \( ( 1 + T^{4} )^{2} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_2$ \( ( 1 + T^{2} )^{4} \)
97$C_2$ \( ( 1 + T^{2} )^{4} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−6.49860957369408700507686464463, −6.42485092721915000073327158869, −6.10943406679652768348230402057, −5.93003974824987697273674769902, −5.88231500710522946659600845852, −5.62668873291195176499693562043, −5.18673344248147815268195993086, −5.04812650819212702495375033823, −5.01393982702516966652878955767, −4.98335495843254955774097140474, −4.48652628543504515649648409540, −4.34156489390107871611728109321, −4.12786225794346189229918619943, −4.04180061169350097558750431173, −3.62453946667223181540704568407, −3.61139618957688211556248202949, −3.36956349871373830768570155578, −2.97165447919623094562908197246, −2.89908403831764470521292838867, −2.38982735964998717767716201328, −1.85630368309975487466113369917, −1.61494921427731080941020762608, −1.14004336220876583025255489716, −1.11190893893154543913427395402, −0.03629652390274289110268692307, 0.03629652390274289110268692307, 1.11190893893154543913427395402, 1.14004336220876583025255489716, 1.61494921427731080941020762608, 1.85630368309975487466113369917, 2.38982735964998717767716201328, 2.89908403831764470521292838867, 2.97165447919623094562908197246, 3.36956349871373830768570155578, 3.61139618957688211556248202949, 3.62453946667223181540704568407, 4.04180061169350097558750431173, 4.12786225794346189229918619943, 4.34156489390107871611728109321, 4.48652628543504515649648409540, 4.98335495843254955774097140474, 5.01393982702516966652878955767, 5.04812650819212702495375033823, 5.18673344248147815268195993086, 5.62668873291195176499693562043, 5.88231500710522946659600845852, 5.93003974824987697273674769902, 6.10943406679652768348230402057, 6.42485092721915000073327158869, 6.49860957369408700507686464463

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