Properties

Label 4-180e2-1.1-c0e2-0-2
Degree $4$
Conductor $32400$
Sign $1$
Analytic cond. $0.00806973$
Root an. cond. $0.299719$
Motivic weight $0$
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-s − 5-s + 6-s − 7-s − 8-s − 10-s − 14-s − 15-s − 16-s − 21-s − 23-s − 24-s − 27-s + 29-s − 30-s + 35-s + 40-s + 41-s − 42-s + 2·43-s − 46-s − 47-s − 48-s + 49-s − 54-s + 56-s + ⋯
L(s)  = 1  + 2-s + 3-s − 5-s + 6-s − 7-s − 8-s − 10-s − 14-s − 15-s − 16-s − 21-s − 23-s − 24-s − 27-s + 29-s − 30-s + 35-s + 40-s + 41-s − 42-s + 2·43-s − 46-s − 47-s − 48-s + 49-s − 54-s + 56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(32400\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(0.00806973\)
Root analytic conductor: \(0.299719\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{180} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 32400,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6674777657\)
\(L(\frac12)\) \(\approx\) \(0.6674777657\)
\(L(1)\) 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$C_2$ \( 1 - T + T^{2} \)
5$C_2$ \( 1 + T + T^{2} \)
good7$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
11$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
13$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
29$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
31$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
43$C_2$ \( ( 1 - T + T^{2} )^{2} \)
47$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
67$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
83$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
89$C_2$ \( ( 1 + T + T^{2} )^{2} \)
97$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
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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

−13.25266003726459056934228925380, −12.63618427465138715563032354413, −12.34233518978514723443508067396, −11.93483636824218202361140453035, −11.32059066798460248258732799065, −10.87563652432827529726951676893, −9.877856881384228442551797346298, −9.755306957471470115556646605831, −9.052683058898347171661999093782, −8.548958346639384954526037770529, −8.175583683876575953362057897275, −7.48473382694494253624140028139, −6.99564495359185224065559830264, −5.96797693585416330644713210577, −5.96541110648486450872095668393, −4.86178503697550767939988741389, −4.02505199161513633549089042307, −3.87921310680456161343166166782, −3.01554335708622184459064059850, −2.52796415098278775375283093093, 2.52796415098278775375283093093, 3.01554335708622184459064059850, 3.87921310680456161343166166782, 4.02505199161513633549089042307, 4.86178503697550767939988741389, 5.96541110648486450872095668393, 5.96797693585416330644713210577, 6.99564495359185224065559830264, 7.48473382694494253624140028139, 8.175583683876575953362057897275, 8.548958346639384954526037770529, 9.052683058898347171661999093782, 9.755306957471470115556646605831, 9.877856881384228442551797346298, 10.87563652432827529726951676893, 11.32059066798460248258732799065, 11.93483636824218202361140453035, 12.34233518978514723443508067396, 12.63618427465138715563032354413, 13.25266003726459056934228925380

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