Properties

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

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s + 9-s − 4·13-s + 4·17-s + 2·25-s + 12·29-s + 12·37-s − 12·41-s − 4·45-s − 14·49-s − 4·53-s − 4·61-s + 16·65-s + 20·73-s + 81-s − 16·85-s − 12·89-s + 4·97-s − 36·101-s − 4·109-s + 36·113-s − 4·117-s − 6·121-s + 28·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1.78·5-s + 1/3·9-s − 1.10·13-s + 0.970·17-s + 2/5·25-s + 2.22·29-s + 1.97·37-s − 1.87·41-s − 0.596·45-s − 2·49-s − 0.549·53-s − 0.512·61-s + 1.98·65-s + 2.34·73-s + 1/9·81-s − 1.73·85-s − 1.27·89-s + 0.406·97-s − 3.58·101-s − 0.383·109-s + 3.38·113-s − 0.369·117-s − 0.545·121-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1152\)    =    \(2^{7} \cdot 3^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1152} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 1152,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.4544183772$
$L(\frac12)$  $\approx$  $0.4544183772$
$L(\frac{3}{2})$   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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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

−14.18198826155077042068868714860, −13.40929752839485230085938675601, −12.32215863397544621963398389319, −12.31725686074733747032880351584, −11.58204746144152243175303945055, −11.02059068705223596291987853205, −9.964671915727173343349228861429, −9.656797755147918818405236990829, −8.173282349834764667604075089133, −8.098990694093691505068092710868, −7.26958585488936859387169650645, −6.42897107072744719896577758454, −5.01357758335939619070346679818, −4.25303028692796488061253338187, −3.14826068230388029805668446658, 3.14826068230388029805668446658, 4.25303028692796488061253338187, 5.01357758335939619070346679818, 6.42897107072744719896577758454, 7.26958585488936859387169650645, 8.098990694093691505068092710868, 8.173282349834764667604075089133, 9.656797755147918818405236990829, 9.964671915727173343349228861429, 11.02059068705223596291987853205, 11.58204746144152243175303945055, 12.31725686074733747032880351584, 12.32215863397544621963398389319, 13.40929752839485230085938675601, 14.18198826155077042068868714860

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