Properties

Degree $4$
Conductor $576$
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  − 2·2-s + 2·4-s − 4·7-s − 9-s + 8·14-s − 4·16-s − 4·17-s + 2·18-s + 8·23-s + 6·25-s − 8·28-s + 4·31-s + 8·32-s + 8·34-s − 2·36-s + 4·41-s − 16·46-s − 24·47-s − 2·49-s − 12·50-s − 8·62-s + 4·63-s − 8·64-s − 8·68-s + 24·71-s − 12·73-s + 20·79-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 1.51·7-s − 1/3·9-s + 2.13·14-s − 16-s − 0.970·17-s + 0.471·18-s + 1.66·23-s + 6/5·25-s − 1.51·28-s + 0.718·31-s + 1.41·32-s + 1.37·34-s − 1/3·36-s + 0.624·41-s − 2.35·46-s − 3.50·47-s − 2/7·49-s − 1.69·50-s − 1.01·62-s + 0.503·63-s − 64-s − 0.970·68-s + 2.84·71-s − 1.40·73-s + 2.25·79-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

Degree: \(4\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{576} (1, \cdot )$
Sato-Tate group: $E_2$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 576,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2239625216\)
\(L(\frac12)\) \(\approx\) \(0.2239625216\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 + p T + p T^{2} \)
3$C_2$ \( 1 + T^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.7054108253, −19.1938156689, −18.8524423811, −18.0069339679, −17.6666951629, −16.8356247312, −16.4912322887, −15.9822387778, −15.2753870091, −14.5735459987, −13.5070874335, −13.0677071494, −12.4305239387, −11.1477625634, −11.0057581452, −9.84857916898, −9.52990583836, −8.76361459870, −8.07692267781, −6.73600798783, −6.63334300780, −4.87562501131, −3.03882077536, 3.03882077536, 4.87562501131, 6.63334300780, 6.73600798783, 8.07692267781, 8.76361459870, 9.52990583836, 9.84857916898, 11.0057581452, 11.1477625634, 12.4305239387, 13.0677071494, 13.5070874335, 14.5735459987, 15.2753870091, 15.9822387778, 16.4912322887, 16.8356247312, 17.6666951629, 18.0069339679, 18.8524423811, 19.1938156689, 19.7054108253

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