Properties

Label 4-180e2-1.1-c0e2-0-1
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 16-s − 25-s − 2·49-s − 4·61-s − 64-s + 100-s + 4·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 2·196-s + 197-s + 199-s + ⋯
L(s)  = 1  − 4-s + 16-s − 25-s − 2·49-s − 4·61-s − 64-s + 100-s + 4·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 2·196-s + 197-s + 199-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

\(L(\frac{1}{2})\) \(\approx\) \(0.3843802773\)
\(L(\frac12)\) \(\approx\) \(0.3843802773\)
\(L(1)\) 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$C_2$ \( 1 + T^{2} \)
3 \( 1 \)
5$C_2$ \( 1 + T^{2} \)
good7$C_2$ \( ( 1 + T^{2} )^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_2$ \( ( 1 + T^{2} )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_2$ \( ( 1 + T^{2} )^{2} \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_2$ \( ( 1 + T^{2} )^{2} \)
43$C_2$ \( ( 1 + T^{2} )^{2} \)
47$C_2$ \( ( 1 + T^{2} )^{2} \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_1$ \( ( 1 + T )^{4} \)
67$C_2$ \( ( 1 + T^{2} )^{2} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_2$ \( ( 1 + T^{2} )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−13.61416047276907475538878389672, −12.68773874131481316135968972332, −12.24786515957334543450342310498, −11.77201509376005350156308605147, −11.12243734817267578579130886343, −10.70856199700068145277697392096, −9.951066944493501011919944634999, −9.768152071362591591297198359402, −9.098082573711693085294217039423, −8.731949602901659574138655814886, −8.002705591916857497167492234651, −7.70984593560360609597004373017, −7.01138108804813254061843343097, −6.00544160009276436343889004620, −5.94207158086714460524284264874, −4.78133270384470936549343138359, −4.65105749693461699670374546515, −3.66834307526012773911986557907, −3.07697035633363634318314595901, −1.72822679618767786561111164747, 1.72822679618767786561111164747, 3.07697035633363634318314595901, 3.66834307526012773911986557907, 4.65105749693461699670374546515, 4.78133270384470936549343138359, 5.94207158086714460524284264874, 6.00544160009276436343889004620, 7.01138108804813254061843343097, 7.70984593560360609597004373017, 8.002705591916857497167492234651, 8.731949602901659574138655814886, 9.098082573711693085294217039423, 9.768152071362591591297198359402, 9.951066944493501011919944634999, 10.70856199700068145277697392096, 11.12243734817267578579130886343, 11.77201509376005350156308605147, 12.24786515957334543450342310498, 12.68773874131481316135968972332, 13.61416047276907475538878389672

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