Properties

Degree 4
Conductor $ 2^{6} \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  − 2·3-s − 4·5-s + 3·9-s + 8·11-s − 4·13-s + 8·15-s + 4·17-s − 8·19-s − 16·23-s + 2·25-s − 4·27-s + 12·29-s + 16·31-s − 16·33-s + 12·37-s + 8·39-s − 12·41-s + 8·43-s − 12·45-s − 14·49-s − 8·51-s − 4·53-s − 32·55-s + 16·57-s + 8·59-s − 4·61-s + 16·65-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.78·5-s + 9-s + 2.41·11-s − 1.10·13-s + 2.06·15-s + 0.970·17-s − 1.83·19-s − 3.33·23-s + 2/5·25-s − 0.769·27-s + 2.22·29-s + 2.87·31-s − 2.78·33-s + 1.97·37-s + 1.28·39-s − 1.87·41-s + 1.21·43-s − 1.78·45-s − 2·49-s − 1.12·51-s − 0.549·53-s − 4.31·55-s + 2.11·57-s + 1.04·59-s − 0.512·61-s + 1.98·65-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(576\)    =    \(2^{6} \cdot 3^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{576} (1, \cdot )$
Sato-Tate  :  $E_1$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 576,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.2906599836$
$L(\frac12)$  $\approx$  $0.2906599836$
$L(\frac{3}{2})$   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$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
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

−19.8934316645, −19.8934316645, −19.1551537949, −19.1551537949, −17.67075195, −17.67075195, −16.6272610009, −16.6272610009, −15.4732516475, −15.4732516475, −14.1819882616, −14.1819882616, −12.3172568607, −12.3172568607, −11.5820474614, −11.5820474614, −9.96467191573, −9.96467191573, −8.09899069409, −8.09899069409, −6.42897107073, −6.42897107073, −4.25303028693, −4.25303028693, 4.25303028693, 4.25303028693, 6.42897107073, 6.42897107073, 8.09899069409, 8.09899069409, 9.96467191573, 9.96467191573, 11.5820474614, 11.5820474614, 12.3172568607, 12.3172568607, 14.1819882616, 14.1819882616, 15.4732516475, 15.4732516475, 16.6272610009, 16.6272610009, 17.67075195, 17.67075195, 19.1551537949, 19.1551537949, 19.8934316645, 19.8934316645

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