Properties

Degree 4
Conductor $ 2^{8} \cdot 3^{3} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s − 8·19-s + 2·25-s + 27-s − 8·43-s + 2·49-s − 8·57-s − 8·67-s + 4·73-s + 2·75-s + 81-s + 4·97-s − 22·121-s + 127-s − 8·129-s + 131-s + 137-s + 139-s + 2·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s − 8·171-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 1.83·19-s + 2/5·25-s + 0.192·27-s − 1.21·43-s + 2/7·49-s − 1.05·57-s − 0.977·67-s + 0.468·73-s + 0.230·75-s + 1/9·81-s + 0.406·97-s − 2·121-s + 0.0887·127-s − 0.704·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.164·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s − 0.611·171-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6912 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6912 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(6912\)    =    \(2^{8} \cdot 3^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6912} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 6912,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.052398998$
$L(\frac12)$  $\approx$  $1.052398998$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 \( 1 \)
3$C_1$ \( 1 - T \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−11.94553134327863302125688168551, −11.33576713873450418581555203819, −10.51465998221254569934018295359, −10.40150764058967059638147587639, −9.474688896854884016708639682987, −8.983018454044175375625782988654, −8.349444577383232947413827186180, −7.937719587681307906842453762127, −7.00058687928287615515234214779, −6.54477445805309970383121722168, −5.70890707132874136144996740956, −4.73962070672136623195764421330, −4.06909507182561132242704751544, −3.07613605045775790107350704794, −1.98816760121008590841728336961, 1.98816760121008590841728336961, 3.07613605045775790107350704794, 4.06909507182561132242704751544, 4.73962070672136623195764421330, 5.70890707132874136144996740956, 6.54477445805309970383121722168, 7.00058687928287615515234214779, 7.937719587681307906842453762127, 8.349444577383232947413827186180, 8.983018454044175375625782988654, 9.474688896854884016708639682987, 10.40150764058967059638147587639, 10.51465998221254569934018295359, 11.33576713873450418581555203819, 11.94553134327863302125688168551

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