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Properties

Label 4-18e2-1.1-c1e2-0-0
Degree $4$
Conductor $324$
Sign $1$
Analytic cond. $0.0206585$
Root an. cond. $0.379118$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3·3-s + 3·6-s − 2·7-s + 8-s + 6·9-s + 3·11-s − 2·13-s + 2·14-s − 16-s − 6·17-s − 6·18-s − 2·19-s + 6·21-s − 3·22-s + 6·23-s − 3·24-s + 5·25-s + 2·26-s − 9·27-s − 6·29-s + 4·31-s − 9·33-s + 6·34-s − 8·37-s + 2·38-s + 6·39-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s + 1.22·6-s − 0.755·7-s + 0.353·8-s + 2·9-s + 0.904·11-s − 0.554·13-s + 0.534·14-s − 1/4·16-s − 1.45·17-s − 1.41·18-s − 0.458·19-s + 1.30·21-s − 0.639·22-s + 1.25·23-s − 0.612·24-s + 25-s + 0.392·26-s − 1.73·27-s − 1.11·29-s + 0.718·31-s − 1.56·33-s + 1.02·34-s − 1.31·37-s + 0.324·38-s + 0.960·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(0.0206585\)
Root analytic conductor: \(0.379118\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 324,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1736174728\)
\(L(\frac12)\) \(\approx\) \(0.1736174728\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
3$C_2$ \( 1 + p T + p T^{2} \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \)
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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

−18.73425098025001598515848722623, −18.65330191638767616820202141858, −17.58588719176622164211034015698, −17.34259594350077650929836362354, −16.65829432309937211505679158926, −16.50555445570130060214767610978, −15.36554110674836471808657271021, −15.04331307163907501027393190909, −13.60256088276205419926735569181, −13.09153056920829555057476441677, −12.18420205922030737509979676105, −11.75914826730233688681995342362, −10.66744030764575546769971244074, −10.51435608625116348572044517930, −9.327762105443371388541282310587, −8.790108040571259251135981778049, −6.92583624002767656428145584125, −6.85650392855044452343093046473, −5.52147912858687978615646302053, −4.36179441556210697223984606647, 4.36179441556210697223984606647, 5.52147912858687978615646302053, 6.85650392855044452343093046473, 6.92583624002767656428145584125, 8.790108040571259251135981778049, 9.327762105443371388541282310587, 10.51435608625116348572044517930, 10.66744030764575546769971244074, 11.75914826730233688681995342362, 12.18420205922030737509979676105, 13.09153056920829555057476441677, 13.60256088276205419926735569181, 15.04331307163907501027393190909, 15.36554110674836471808657271021, 16.50555445570130060214767610978, 16.65829432309937211505679158926, 17.34259594350077650929836362354, 17.58588719176622164211034015698, 18.65330191638767616820202141858, 18.73425098025001598515848722623

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