Properties

Label 4-1200e2-1.1-c0e2-0-2
Degree $4$
Conductor $1440000$
Sign $1$
Analytic cond. $0.358654$
Root an. cond. $0.773872$
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  − 2·3-s − 4-s + 3·9-s + 2·12-s + 16-s + 2·17-s + 2·19-s − 2·23-s − 4·27-s − 3·36-s + 2·47-s − 2·48-s − 4·51-s + 4·53-s − 4·57-s − 2·61-s − 64-s − 2·68-s + 4·69-s − 2·76-s + 5·81-s + 2·92-s + 4·108-s − 2·109-s + 2·113-s + 127-s + 131-s + ⋯
L(s)  = 1  − 2·3-s − 4-s + 3·9-s + 2·12-s + 16-s + 2·17-s + 2·19-s − 2·23-s − 4·27-s − 3·36-s + 2·47-s − 2·48-s − 4·51-s + 4·53-s − 4·57-s − 2·61-s − 64-s − 2·68-s + 4·69-s − 2·76-s + 5·81-s + 2·92-s + 4·108-s − 2·109-s + 2·113-s + 127-s + 131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1440000\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(0.358654\)
Root analytic conductor: \(0.773872\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1200} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1440000,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4788017927\)
\(L(\frac12)\) \(\approx\) \(0.4788017927\)
\(L(1)\) 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$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
good7$C_2^2$ \( 1 + T^{4} \)
11$C_2^2$ \( 1 + T^{4} \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
17$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
19$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
23$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
29$C_2^2$ \( 1 + T^{4} \)
31$C_2$ \( ( 1 + T^{2} )^{2} \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
41$C_2$ \( ( 1 + T^{2} )^{2} \)
43$C_2$ \( ( 1 + T^{2} )^{2} \)
47$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
53$C_1$ \( ( 1 - T )^{4} \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
67$C_2$ \( ( 1 + T^{2} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2^2$ \( 1 + T^{4} \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_2^2$ \( 1 + T^{4} \)
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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

−10.12738885703313631899197824065, −9.934631146403040635609845463874, −9.348439253969716103499452070385, −9.342109064604868001304213804416, −8.397423941846263771717808358874, −8.059974699900014531791475717678, −7.42159083927956111643318141993, −7.39631564949780711285228755408, −6.92886345947696367569402693165, −6.02310815220676841352582560109, −5.80669513636148049916062697368, −5.44858057555189492055433108603, −5.44554118987709071019299583272, −4.45815436208753662970914316506, −4.42135424615880239583689128040, −3.60230965685319239762687001590, −3.44437852206595666559099523709, −2.25868528192296967182604056390, −1.25506498967479330315190931877, −0.861272831468017989123779469564, 0.861272831468017989123779469564, 1.25506498967479330315190931877, 2.25868528192296967182604056390, 3.44437852206595666559099523709, 3.60230965685319239762687001590, 4.42135424615880239583689128040, 4.45815436208753662970914316506, 5.44554118987709071019299583272, 5.44858057555189492055433108603, 5.80669513636148049916062697368, 6.02310815220676841352582560109, 6.92886345947696367569402693165, 7.39631564949780711285228755408, 7.42159083927956111643318141993, 8.059974699900014531791475717678, 8.397423941846263771717808358874, 9.342109064604868001304213804416, 9.348439253969716103499452070385, 9.934631146403040635609845463874, 10.12738885703313631899197824065

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