Properties

Degree 4
Conductor $ 3^{4} \cdot 13^{2} $
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  − 2·7-s − 16-s − 2·19-s + 2·31-s + 2·37-s + 2·49-s − 2·67-s − 2·73-s + 2·97-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 + 191-s + 193-s + ⋯
L(s)  = 1  − 2·7-s − 16-s − 2·19-s + 2·31-s + 2·37-s + 2·49-s − 2·67-s − 2·73-s + 2·97-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 + 191-s + 193-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(13689\)    =    \(3^{4} \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{117} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 13689,\ (\ :0, 0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.3101862578$
$L(\frac12)$  $\approx$  $0.3101862578$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;13\}$, \(F_p\) is a polynomial of degree 4. If $p \in \{3,\;13\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3 \( 1 \)
13$C_2$ \( 1 + T^{2} \)
good2$C_2^2$ \( 1 + T^{4} \)
5$C_2^2$ \( 1 + T^{4} \)
7$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
11$C_2^2$ \( 1 + T^{4} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
37$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
41$C_2^2$ \( 1 + T^{4} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
71$C_2^2$ \( 1 + T^{4} \)
73$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_2^2$ \( 1 + T^{4} \)
97$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−13.80990494819017442544207341531, −13.36001136281506947321561544962, −13.05270677275964737188948525381, −12.72164914596868163355269831842, −11.93499686341665259569750108108, −11.67036785348349858475122944719, −10.65262963256653539214358760704, −10.52450589542211795910627174439, −9.701049372459152198528131483308, −9.460599077616326652932631890719, −8.744582829751551591102352480621, −8.287424301189132676813928040431, −7.37991611206024721121675636931, −6.74227370788945589021286031122, −6.18811586118836933892295665132, −6.02436325627031227717949276801, −4.57945296969333583056588158114, −4.21585912692895745454493259751, −3.10156635655119368499674695018, −2.45453865405386386796231817165, 2.45453865405386386796231817165, 3.10156635655119368499674695018, 4.21585912692895745454493259751, 4.57945296969333583056588158114, 6.02436325627031227717949276801, 6.18811586118836933892295665132, 6.74227370788945589021286031122, 7.37991611206024721121675636931, 8.287424301189132676813928040431, 8.744582829751551591102352480621, 9.460599077616326652932631890719, 9.701049372459152198528131483308, 10.52450589542211795910627174439, 10.65262963256653539214358760704, 11.67036785348349858475122944719, 11.93499686341665259569750108108, 12.72164914596868163355269831842, 13.05270677275964737188948525381, 13.36001136281506947321561544962, 13.80990494819017442544207341531

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