Properties

Label 4-40e4-1.1-c0e2-0-0
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·7-s + 2·9-s + 4·21-s + 2·23-s − 2·27-s + 2·43-s + 2·47-s + 2·49-s − 4·63-s + 2·67-s − 4·69-s + 3·81-s + 2·83-s − 4·101-s − 2·103-s − 2·107-s − 2·121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s − 4·141-s − 4·147-s + 149-s + 151-s + ⋯
L(s)  = 1  − 2·3-s − 2·7-s + 2·9-s + 4·21-s + 2·23-s − 2·27-s + 2·43-s + 2·47-s + 2·49-s − 4·63-s + 2·67-s − 4·69-s + 3·81-s + 2·83-s − 4·101-s − 2·103-s − 2·107-s − 2·121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s − 4·141-s − 4·147-s + 149-s + 151-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

\(L(\frac{1}{2})\) \(\approx\) \(0.3497200583\)
\(L(\frac12)\) \(\approx\) \(0.3497200583\)
\(L(1)\) 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_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
7$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
11$C_2$ \( ( 1 + T^{2} )^{2} \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_2^2$ \( 1 + T^{4} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{2} \)
37$C_2^2$ \( 1 + T^{4} \)
41$C_2$ \( ( 1 + T^{2} )^{2} \)
43$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
47$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
53$C_2^2$ \( 1 + T^{4} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_2^2$ \( 1 + T^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
89$C_2$ \( ( 1 + T^{2} )^{2} \)
97$C_2^2$ \( 1 + T^{4} \)
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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

−9.729460302158698731675019460483, −9.466791997473487130280737475219, −9.060901116853319297913509992627, −9.043928265447382677534423569293, −7.917833732876467541241229579751, −7.85505237748298831364543957271, −6.95941736209400434148230776282, −6.92340797262602558140441530626, −6.55986998524535030237384893101, −6.22041960582588081252379114895, −5.60929089008795643751488387041, −5.43784671243576231369934894367, −5.18714801062533176031294069748, −4.36706039008445730316710681308, −3.94599839881235936888079873317, −3.58261539079594797861871942478, −2.70410768916376011757813084711, −2.56395168842232300890867732186, −1.24958978499490677720461180141, −0.59147959482019037054521395577, 0.59147959482019037054521395577, 1.24958978499490677720461180141, 2.56395168842232300890867732186, 2.70410768916376011757813084711, 3.58261539079594797861871942478, 3.94599839881235936888079873317, 4.36706039008445730316710681308, 5.18714801062533176031294069748, 5.43784671243576231369934894367, 5.60929089008795643751488387041, 6.22041960582588081252379114895, 6.55986998524535030237384893101, 6.92340797262602558140441530626, 6.95941736209400434148230776282, 7.85505237748298831364543957271, 7.917833732876467541241229579751, 9.043928265447382677534423569293, 9.060901116853319297913509992627, 9.466791997473487130280737475219, 9.729460302158698731675019460483

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