Properties

Degree 4
Conductor $ 2^{8} \cdot 3^{6} \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s + 4·7-s + 3·11-s − 5·13-s + 3·17-s + 5·19-s + 3·23-s + 5·25-s − 3·29-s + 8·31-s − 12·35-s + 7·37-s − 9·41-s + 11·43-s + 9·49-s − 3·53-s − 9·55-s + 24·59-s + 4·61-s + 15·65-s + 8·67-s − 11·73-s + 12·77-s − 16·79-s − 3·83-s − 9·85-s + 15·89-s + ⋯
L(s)  = 1  − 1.34·5-s + 1.51·7-s + 0.904·11-s − 1.38·13-s + 0.727·17-s + 1.14·19-s + 0.625·23-s + 25-s − 0.557·29-s + 1.43·31-s − 2.02·35-s + 1.15·37-s − 1.40·41-s + 1.67·43-s + 9/7·49-s − 0.412·53-s − 1.21·55-s + 3.12·59-s + 0.512·61-s + 1.86·65-s + 0.977·67-s − 1.28·73-s + 1.36·77-s − 1.80·79-s − 0.329·83-s − 0.976·85-s + 1.58·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(9144576\)    =    \(2^{8} \cdot 3^{6} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{3024} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 9144576,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(3.189497559\)
\(L(\frac12)\)  \(\approx\)  \(3.189497559\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. 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 \( 1 \)
3 \( 1 \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 11 T + 78 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 3 T - 74 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} \)
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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

−8.691703799719210854682480510376, −8.560354481034445039471003182280, −8.042413110337622028823033812340, −7.84063556711123871832434234637, −7.35960431667286084510556057920, −7.22049715118263417289468008044, −6.92477729801594174902433456730, −6.31173873539867523202513171634, −5.65101826124369984931919018617, −5.48079937631410340763345740791, −4.88645056213009144906849478884, −4.67639345506329846383875361261, −4.22249335869452173749016267614, −3.95255198246004845921944366070, −3.22912804504060757296101278912, −3.02740724669832917063151796102, −2.27675496325682002985285286244, −1.81911240424870450360908710073, −0.872358056979913534653660343269, −0.806487909841070155741251105632, 0.806487909841070155741251105632, 0.872358056979913534653660343269, 1.81911240424870450360908710073, 2.27675496325682002985285286244, 3.02740724669832917063151796102, 3.22912804504060757296101278912, 3.95255198246004845921944366070, 4.22249335869452173749016267614, 4.67639345506329846383875361261, 4.88645056213009144906849478884, 5.48079937631410340763345740791, 5.65101826124369984931919018617, 6.31173873539867523202513171634, 6.92477729801594174902433456730, 7.22049715118263417289468008044, 7.35960431667286084510556057920, 7.84063556711123871832434234637, 8.042413110337622028823033812340, 8.560354481034445039471003182280, 8.691703799719210854682480510376

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