Properties

Label 4-1440e2-1.1-c0e2-0-0
Degree $4$
Conductor $2073600$
Sign $1$
Analytic cond. $0.516463$
Root an. cond. $0.847734$
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  − 2·13-s + 2·17-s − 25-s + 2·37-s − 2·53-s + 2·73-s + 2·97-s + 4·101-s + 2·113-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 + 197-s + 199-s + ⋯
L(s)  = 1  − 2·13-s + 2·17-s − 25-s + 2·37-s − 2·53-s + 2·73-s + 2·97-s + 4·101-s + 2·113-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 + 197-s + 199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2073600\)    =    \(2^{10} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(0.516463\)
Root analytic conductor: \(0.847734\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2073600,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.007171973\)
\(L(\frac12)\) \(\approx\) \(1.007171973\)
\(L(1)\) 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 \( 1 \)
3 \( 1 \)
5$C_2$ \( 1 + T^{2} \)
good7$C_2^2$ \( 1 + T^{4} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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$C_2$ \( ( 1 + T^{2} )^{2} \)
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

−9.948291209855228656984027196396, −9.619135127605727888345857042665, −9.268509675723928818072588730068, −8.805170338517886325761611565369, −8.120543968661658975165355533118, −7.81194292611428217007354855222, −7.47779700011647836387520347388, −7.45672434476210826111335352555, −6.63557196822911011547672148525, −6.10480684256356854509112507007, −5.96945497089711161558334807659, −5.26430892208687826208643730303, −4.83510087779419007068133314720, −4.70980916418435184764549695298, −3.83987950428104676100348833714, −3.48938460250877756966617029190, −2.88015613929687152014350821595, −2.38113606625636665596598092352, −1.80431104230119115090236852145, −0.848296586500977528890620058934, 0.848296586500977528890620058934, 1.80431104230119115090236852145, 2.38113606625636665596598092352, 2.88015613929687152014350821595, 3.48938460250877756966617029190, 3.83987950428104676100348833714, 4.70980916418435184764549695298, 4.83510087779419007068133314720, 5.26430892208687826208643730303, 5.96945497089711161558334807659, 6.10480684256356854509112507007, 6.63557196822911011547672148525, 7.45672434476210826111335352555, 7.47779700011647836387520347388, 7.81194292611428217007354855222, 8.120543968661658975165355533118, 8.805170338517886325761611565369, 9.268509675723928818072588730068, 9.619135127605727888345857042665, 9.948291209855228656984027196396

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