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Properties

Label 4-40e4-1.1-c0e2-0-2
Degree $4$
Conductor $2560000$
Sign $1$
Analytic cond. $0.637608$
Root an. cond. $0.893590$
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 + 2·37-s + 2·53-s − 2·73-s − 81-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 + 2·37-s + 2·53-s − 2·73-s − 81-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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2560000\)    =    \(2^{12} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(0.637608\)
Root analytic conductor: \(0.893590\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2560000,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.080586333\)
\(L(\frac12)\) \(\approx\) \(1.080586333\)
\(L(1)\) 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 \( 1 \)
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_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

−9.961734970081460408075153894472, −9.548399693725478716586461596171, −8.876564417090761194257583351309, −8.798436193317607234406669320793, −8.091247731466074846928739283111, −7.66579803539722451925421093393, −7.41883908849089870645206731706, −7.29226262087773670281017124505, −6.56712806286335577060941872781, −6.13321397127507000793770628340, −5.56364618365182883556818772206, −5.44700313966039662178670357305, −4.78836929313519103417948617163, −4.48220157498074805728549511348, −3.90170104291048891170229486744, −3.36643280550921688563021665879, −2.64759275157110111882515957514, −2.58330657180352164610614307723, −1.66118871882881372317107720211, −0.860198330781881761685279441821, 0.860198330781881761685279441821, 1.66118871882881372317107720211, 2.58330657180352164610614307723, 2.64759275157110111882515957514, 3.36643280550921688563021665879, 3.90170104291048891170229486744, 4.48220157498074805728549511348, 4.78836929313519103417948617163, 5.44700313966039662178670357305, 5.56364618365182883556818772206, 6.13321397127507000793770628340, 6.56712806286335577060941872781, 7.29226262087773670281017124505, 7.41883908849089870645206731706, 7.66579803539722451925421093393, 8.091247731466074846928739283111, 8.798436193317607234406669320793, 8.876564417090761194257583351309, 9.548399693725478716586461596171, 9.961734970081460408075153894472

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