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 − 2.79·5-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 + 14/5·80-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  =  \((4,\ 11664,\ (\ :1, 1),\ 1)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.38982\)
\(L(\frac12)\)  \(\approx\)  \(1.38982\)
\(L(2)\)   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

−14.11159480552858193627565991505, −13.09096465422954007633459593371, −12.70382142159322865412156595753, −12.12908326588426302492827968801, −11.74036510672811192097502507037, −11.14844941353113461753741700841, −11.10284807026164305307958769683, −10.27169228458798683966810714111, −9.089151567722652576589321718308, −8.683447153555869748696341533782, −8.318129157824399392637293886953, −7.57689051922074984065029873599, −7.25391532382315840977502948194, −5.94959904048751100675937442017, −5.94677577327664609242353531672, −4.58790222975333020474746093491, −4.09671298587819528734407371992, −3.57264637919876430099973986581, −3.28938179994876724343884246102, −0.801805983683099276174956392120, 0.801805983683099276174956392120, 3.28938179994876724343884246102, 3.57264637919876430099973986581, 4.09671298587819528734407371992, 4.58790222975333020474746093491, 5.94677577327664609242353531672, 5.94959904048751100675937442017, 7.25391532382315840977502948194, 7.57689051922074984065029873599, 8.318129157824399392637293886953, 8.683447153555869748696341533782, 9.089151567722652576589321718308, 10.27169228458798683966810714111, 11.10284807026164305307958769683, 11.14844941353113461753741700841, 11.74036510672811192097502507037, 12.12908326588426302492827968801, 12.70382142159322865412156595753, 13.09096465422954007633459593371, 14.11159480552858193627565991505

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