Properties

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

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 4·7-s − 5·11-s + 5·13-s + 3·17-s + 19-s − 3·23-s + 5·25-s − 29-s + 4·35-s − 3·37-s − 5·41-s − 43-s + 9·49-s − 9·53-s + 5·55-s − 28·61-s − 5·65-s − 8·67-s − 24·71-s − 3·73-s + 20·77-s − 16·79-s + 9·83-s − 3·85-s − 13·89-s − 20·91-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.51·7-s − 1.50·11-s + 1.38·13-s + 0.727·17-s + 0.229·19-s − 0.625·23-s + 25-s − 0.185·29-s + 0.676·35-s − 0.493·37-s − 0.780·41-s − 0.152·43-s + 9/7·49-s − 1.23·53-s + 0.674·55-s − 3.58·61-s − 0.620·65-s − 0.977·67-s − 2.84·71-s − 0.351·73-s + 2.27·77-s − 1.80·79-s + 0.987·83-s − 0.325·85-s − 1.37·89-s − 2.09·91-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  =  \(2\)
Selberg data  =  \((4,\ 9144576,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   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$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 3 T - 64 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 13 T + 80 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 9 T - 16 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
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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

−8.600423653229840408808767033313, −8.089451762921215009046955054423, −7.72658652821891308715503167176, −7.56980103187178263220904642885, −6.93528039698930848788789197911, −6.76692288238575305659273914060, −6.08773006047263752108762735136, −5.94465366090596948038857765762, −5.64881878332391680566977770496, −5.10467047115432164762679956830, −4.43599268181982527574138730490, −4.39143190083494020368950107795, −3.49376852939387318069389909272, −3.31691634694299195356058096122, −2.97913221524824568441031630067, −2.62467224757714468248644558341, −1.61731925739972133106433759324, −1.26860482922731089060873685054, 0, 0, 1.26860482922731089060873685054, 1.61731925739972133106433759324, 2.62467224757714468248644558341, 2.97913221524824568441031630067, 3.31691634694299195356058096122, 3.49376852939387318069389909272, 4.39143190083494020368950107795, 4.43599268181982527574138730490, 5.10467047115432164762679956830, 5.64881878332391680566977770496, 5.94465366090596948038857765762, 6.08773006047263752108762735136, 6.76692288238575305659273914060, 6.93528039698930848788789197911, 7.56980103187178263220904642885, 7.72658652821891308715503167176, 8.089451762921215009046955054423, 8.600423653229840408808767033313

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