Properties

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

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 14·5-s + 8·8-s − 28·10-s + 40·13-s − 16·16-s − 16·17-s + 97·25-s − 80·26-s + 20·29-s + 32·34-s − 20·37-s + 112·40-s − 100·41-s + 23·49-s − 194·50-s − 94·53-s − 40·58-s − 128·61-s + 64·64-s + 560·65-s − 110·73-s + 40·74-s − 224·80-s + 200·82-s − 224·85-s + 20·89-s + ⋯
L(s)  = 1  − 2-s + 14/5·5-s + 8-s − 2.79·10-s + 3.07·13-s − 16-s − 0.941·17-s + 3.87·25-s − 3.07·26-s + 0.689·29-s + 0.941·34-s − 0.540·37-s + 14/5·40-s − 2.43·41-s + 0.469·49-s − 3.87·50-s − 1.77·53-s − 0.689·58-s − 2.09·61-s + 64-s + 8.61·65-s − 1.50·73-s + 0.540·74-s − 2.79·80-s + 2.43·82-s − 2.63·85-s + 0.224·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11664 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(11664\)    =    \(2^{4} \cdot 3^{6}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{108} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 11664,\ (\ :1, 1),\ 1)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.63617\)
\(L(\frac12)\)  \(\approx\)  \(1.63617\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 4. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 + p T + p^{2} T^{2} \)
3 \( 1 \)
good5$C_2$ \( ( 1 - 7 T + p^{2} T^{2} )^{2} \)
7$C_2$ \( ( 1 - 11 T + p^{2} T^{2} )( 1 + 11 T + p^{2} T^{2} ) \)
11$C_2^2$ \( 1 - 167 T^{2} + p^{4} T^{4} \)
13$C_2$ \( ( 1 - 20 T + p^{2} T^{2} )^{2} \)
17$C_2$ \( ( 1 + 8 T + p^{2} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 614 T^{2} + p^{4} T^{4} \)
23$C_2^2$ \( 1 - 1046 T^{2} + p^{4} T^{4} \)
29$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )^{2} \)
31$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
37$C_2$ \( ( 1 + 10 T + p^{2} T^{2} )^{2} \)
41$C_2$ \( ( 1 + 50 T + p^{2} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 3398 T^{2} + p^{4} T^{4} \)
47$C_2^2$ \( 1 + 3082 T^{2} + p^{4} T^{4} \)
53$C_2$ \( ( 1 + 47 T + p^{2} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 5762 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 + 64 T + p^{2} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 1478 T^{2} + p^{4} T^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
73$C_2$ \( ( 1 + 55 T + p^{2} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 12434 T^{2} + p^{4} T^{4} \)
83$C_2^2$ \( 1 - 12911 T^{2} + p^{4} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )^{2} \)
97$C_2$ \( ( 1 + 25 T + p^{2} T^{2} )^{2} \)
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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

−13.65171717859994979819767824748, −13.41405183791542871506647639331, −13.11240047239666905369613040605, −12.24432476250521832099858364681, −11.15001896174840744393383480555, −10.95749229245044362711917195548, −10.23232057975926124365667829440, −10.12899020510223067369966124106, −9.323127331205507395306825099192, −8.919945722914774266455946141385, −8.654477570599061711113866632041, −8.003623767483553983288030209150, −6.72125585241181109642264742912, −6.45257186795273421476901028968, −5.88395037204140374377020683437, −5.28648611455279355077013166041, −4.33574628821866167767952593205, −3.14448217799822426958685406920, −1.72898840209495218283428661696, −1.48497583204864638617788270488, 1.48497583204864638617788270488, 1.72898840209495218283428661696, 3.14448217799822426958685406920, 4.33574628821866167767952593205, 5.28648611455279355077013166041, 5.88395037204140374377020683437, 6.45257186795273421476901028968, 6.72125585241181109642264742912, 8.003623767483553983288030209150, 8.654477570599061711113866632041, 8.919945722914774266455946141385, 9.323127331205507395306825099192, 10.12899020510223067369966124106, 10.23232057975926124365667829440, 10.95749229245044362711917195548, 11.15001896174840744393383480555, 12.24432476250521832099858364681, 13.11240047239666905369613040605, 13.41405183791542871506647639331, 13.65171717859994979819767824748

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