Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{4} \cdot 7^{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  + 2·2-s + 3·4-s + 4·8-s + 5·16-s − 4·29-s + 6·32-s − 8·58-s + 7·64-s − 12·116-s + 2·121-s + 127-s + 8·128-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 + 197-s + 199-s + ⋯
L(s)  = 1  + 2·2-s + 3·4-s + 4·8-s + 5·16-s − 4·29-s + 6·32-s − 8·58-s + 7·64-s − 12·116-s + 2·121-s + 127-s + 8·128-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 + 197-s + 199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3111696\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{1764} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 3111696,\ (\ :0, 0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(4.594417780\)
\(L(\frac12)\)  \(\approx\)  \(4.594417780\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

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

Imaginary part of the first few zeros on the critical line

−9.659383482076321123103911949970, −9.507238998594422910190845815944, −8.845207762284210828230873090792, −8.409820620381303945906461342835, −7.71028414437749712951653289369, −7.60420758735182749836449887121, −7.22004980049350719028832459650, −6.86504228984014016252219503405, −6.18460956923895546934513110706, −6.07407141785245440799488904403, −5.46804349956058881904074425448, −5.31451908067224196334220458268, −4.79943829362728001219307872881, −4.25645906281573239712506438275, −3.74845511281811893891380091675, −3.63332251567808974695681433885, −3.01196616588470107968316419511, −2.36291571661188020919983003193, −1.94540061496984898281872094842, −1.39157937932907991579790493048, 1.39157937932907991579790493048, 1.94540061496984898281872094842, 2.36291571661188020919983003193, 3.01196616588470107968316419511, 3.63332251567808974695681433885, 3.74845511281811893891380091675, 4.25645906281573239712506438275, 4.79943829362728001219307872881, 5.31451908067224196334220458268, 5.46804349956058881904074425448, 6.07407141785245440799488904403, 6.18460956923895546934513110706, 6.86504228984014016252219503405, 7.22004980049350719028832459650, 7.60420758735182749836449887121, 7.71028414437749712951653289369, 8.409820620381303945906461342835, 8.845207762284210828230873090792, 9.507238998594422910190845815944, 9.659383482076321123103911949970

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