Properties

Label 4-1280e2-1.1-c0e2-0-7
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

\(L(\frac{1}{2})\) \(\approx\) \(1.639538085\)
\(L(\frac12)\) \(\approx\) \(1.639538085\)
\(L(1)\) 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

−10.07403066264750029921991646453, −9.709306267011088429070222059352, −9.017766454665096541390124675676, −8.947732681856564842015412916729, −8.590450849705363797826252254669, −8.438466821913513519585770745763, −7.49173514819837414169334955315, −7.04457857340801992451861750460, −6.58780435151141343806501754820, −6.42925465942160716011031093663, −5.79187168655000644734836805734, −5.74474983610210005102651214319, −4.97789828717673226204301224537, −4.75463737771097494376872630462, −3.80848993559041850282377922754, −3.74021845670869840604791606181, −2.57835656525677766203580935029, −2.56382699017365319480480876623, −1.59728579618910801388038849945, −1.37436779908331870438816338532, 1.37436779908331870438816338532, 1.59728579618910801388038849945, 2.56382699017365319480480876623, 2.57835656525677766203580935029, 3.74021845670869840604791606181, 3.80848993559041850282377922754, 4.75463737771097494376872630462, 4.97789828717673226204301224537, 5.74474983610210005102651214319, 5.79187168655000644734836805734, 6.42925465942160716011031093663, 6.58780435151141343806501754820, 7.04457857340801992451861750460, 7.49173514819837414169334955315, 8.438466821913513519585770745763, 8.590450849705363797826252254669, 8.947732681856564842015412916729, 9.017766454665096541390124675676, 9.709306267011088429070222059352, 10.07403066264750029921991646453

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