Properties

Label 4-1280e2-1.1-c0e2-0-0
Degree $4$
Conductor $1638400$
Sign $1$
Analytic cond. $0.408069$
Root an. cond. $0.799251$
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  − 2·5-s − 2·13-s − 2·17-s + 3·25-s + 2·37-s − 2·53-s + 4·65-s − 2·73-s − 81-s + 4·85-s + 2·97-s + 4·109-s − 2·113-s + 2·121-s − 4·125-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 + ⋯
L(s)  = 1  − 2·5-s − 2·13-s − 2·17-s + 3·25-s + 2·37-s − 2·53-s + 4·65-s − 2·73-s − 81-s + 4·85-s + 2·97-s + 4·109-s − 2·113-s + 2·121-s − 4·125-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1638400\)    =    \(2^{16} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(0.408069\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1638400,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3950857275\)
\(L(\frac12)\) \(\approx\) \(0.3950857275\)
\(L(1)\) 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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 + T^{4} \)
7$C_2^2$ \( 1 + T^{4} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
17$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
19$C_2$ \( ( 1 + T^{2} )^{2} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{2} \)
37$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
41$C_2$ \( ( 1 + T^{2} )^{2} \)
43$C_2^2$ \( 1 + T^{4} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
59$C_2$ \( ( 1 + T^{2} )^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
67$C_2^2$ \( 1 + T^{4} \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
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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

−10.17838798312829118638460205802, −9.549043106157519284353947974808, −9.194119156098533152783828734163, −8.862133571389173939098725474424, −8.390080089110390140436706790188, −7.925364502967568629946506837545, −7.67555328746532838095396762153, −7.23804426260962121104767848944, −6.98510380780720997632179167339, −6.49651926228537385469881830752, −6.01829734414518287410648994780, −5.30401274665158675155824955763, −4.58972446652015617176234769491, −4.45135557577723848392513528084, −4.43813509290071215277046417838, −3.48790581430571638952733573985, −2.99792808947332476694819824521, −2.55676696881410933695317694536, −1.86337020966777733282503932303, −0.52541391397976017058311200122, 0.52541391397976017058311200122, 1.86337020966777733282503932303, 2.55676696881410933695317694536, 2.99792808947332476694819824521, 3.48790581430571638952733573985, 4.43813509290071215277046417838, 4.45135557577723848392513528084, 4.58972446652015617176234769491, 5.30401274665158675155824955763, 6.01829734414518287410648994780, 6.49651926228537385469881830752, 6.98510380780720997632179167339, 7.23804426260962121104767848944, 7.67555328746532838095396762153, 7.925364502967568629946506837545, 8.390080089110390140436706790188, 8.862133571389173939098725474424, 9.194119156098533152783828734163, 9.549043106157519284353947974808, 10.17838798312829118638460205802

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