Properties

Label 4-1280e2-1.1-c0e2-0-3
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 − 4·61-s − 4·65-s + 2·73-s − 81-s − 4·85-s − 2·97-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 − 4·61-s − 4·65-s + 2·73-s − 81-s − 4·85-s − 2·97-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

\(L(\frac{1}{2})\) \(\approx\) \(0.8872244783\)
\(L(\frac12)\) \(\approx\) \(0.8872244783\)
\(L(1)\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_1$ \( ( 1 + T )^{4} \)
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

−10.23711400181185350561866138118, −9.541016757058697470036630218576, −9.118745913911783528720798821165, −8.886505133414054586252532406794, −8.146170534884785783734466792720, −8.092599551787920567021759210917, −7.77302220976282051780413503769, −7.42542116703009233902558842530, −6.73332600072160857764650719305, −6.51986457341453147936401563017, −5.74152591236722501722998041618, −5.65653986091227342457376095456, −4.87939390352804432372898037592, −4.36930217022453993242359497317, −3.84261966850104562744588414904, −3.75065756697698574979972435297, −3.03584198863477598145321032218, −2.77534599961601714499993196527, −1.37751409789178852066729876465, −0.983393780236680096951006918568, 0.983393780236680096951006918568, 1.37751409789178852066729876465, 2.77534599961601714499993196527, 3.03584198863477598145321032218, 3.75065756697698574979972435297, 3.84261966850104562744588414904, 4.36930217022453993242359497317, 4.87939390352804432372898037592, 5.65653986091227342457376095456, 5.74152591236722501722998041618, 6.51986457341453147936401563017, 6.73332600072160857764650719305, 7.42542116703009233902558842530, 7.77302220976282051780413503769, 8.092599551787920567021759210917, 8.146170534884785783734466792720, 8.886505133414054586252532406794, 9.118745913911783528720798821165, 9.541016757058697470036630218576, 10.23711400181185350561866138118

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