Properties

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

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3·4-s − 2·7-s + 9-s − 3·12-s − 4·13-s + 5·16-s + 8·19-s − 2·21-s − 6·25-s + 27-s + 6·28-s − 3·36-s + 12·37-s − 4·39-s − 8·43-s + 5·48-s + 3·49-s + 12·52-s + 8·57-s − 4·61-s − 2·63-s − 3·64-s + 8·67-s − 12·73-s − 6·75-s − 24·76-s + ⋯
L(s)  = 1  + 0.577·3-s − 3/2·4-s − 0.755·7-s + 1/3·9-s − 0.866·12-s − 1.10·13-s + 5/4·16-s + 1.83·19-s − 0.436·21-s − 6/5·25-s + 0.192·27-s + 1.13·28-s − 1/2·36-s + 1.97·37-s − 0.640·39-s − 1.21·43-s + 0.721·48-s + 3/7·49-s + 1.66·52-s + 1.05·57-s − 0.512·61-s − 0.251·63-s − 3/8·64-s + 0.977·67-s − 1.40·73-s − 0.692·75-s − 2.75·76-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1323\)    =    \(3^{3} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1323} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 1323,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.4977203473$
$L(\frac12)$  $\approx$  $0.4977203473$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;7\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_1$ \( 1 - T \)
7$C_1$ \( ( 1 + T )^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
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$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
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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

−13.77909280462587463511606590839, −13.19379250527463570089741680145, −13.00543957503168228446980436667, −12.02428486441375050992152755294, −11.55595081754559805664353277943, −10.18331996404873608095625884352, −9.724661210726173973340366981146, −9.457705101580370517739336150901, −8.672365196683144779961098572501, −7.81295430367747747098376616892, −7.25047783802838427330028450541, −5.94607665445287604157300008429, −5.02709416296405066539370067618, −4.13559084050773741974089362056, −3.05422074105458389777226041971, 3.05422074105458389777226041971, 4.13559084050773741974089362056, 5.02709416296405066539370067618, 5.94607665445287604157300008429, 7.25047783802838427330028450541, 7.81295430367747747098376616892, 8.672365196683144779961098572501, 9.457705101580370517739336150901, 9.724661210726173973340366981146, 10.18331996404873608095625884352, 11.55595081754559805664353277943, 12.02428486441375050992152755294, 13.00543957503168228446980436667, 13.19379250527463570089741680145, 13.77909280462587463511606590839

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