Properties

Label 4-40e4-1.1-c1e2-0-40
Degree $4$
Conductor $2560000$
Sign $1$
Analytic cond. $163.227$
Root an. cond. $3.57436$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4·7-s + 2·9-s − 2·11-s + 2·13-s + 6·19-s + 8·21-s − 12·23-s − 6·27-s − 6·29-s + 16·31-s + 4·33-s − 6·37-s − 4·39-s − 10·43-s − 2·49-s − 10·53-s − 12·57-s − 6·59-s − 18·61-s − 8·63-s + 10·67-s + 24·69-s − 8·73-s + 8·77-s + 11·81-s − 2·83-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.51·7-s + 2/3·9-s − 0.603·11-s + 0.554·13-s + 1.37·19-s + 1.74·21-s − 2.50·23-s − 1.15·27-s − 1.11·29-s + 2.87·31-s + 0.696·33-s − 0.986·37-s − 0.640·39-s − 1.52·43-s − 2/7·49-s − 1.37·53-s − 1.58·57-s − 0.781·59-s − 2.30·61-s − 1.00·63-s + 1.22·67-s + 2.88·69-s − 0.936·73-s + 0.911·77-s + 11/9·81-s − 0.219·83-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(2560000\)    =    \(2^{12} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(163.227\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2560000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−9.409849107021290924212913667598, −8.973240765902173635808131042840, −8.166221422442874923142527804299, −8.021975407540844688834874989070, −7.69637422811741651850934558385, −7.10033155884986463762739968618, −6.47840116003058930629506805179, −6.42877478120596479587860185234, −5.87631680476208146052773780967, −5.82865531445608805589229308253, −5.00401420807203998386668496800, −4.86788396244498416072826967746, −4.09648511318576568001315043004, −3.66254060472490055928832640638, −3.18816530161154160413273479350, −2.77741016294620158090345512097, −1.87310610378703042054930399652, −1.28971217048944487230143332373, 0, 0, 1.28971217048944487230143332373, 1.87310610378703042054930399652, 2.77741016294620158090345512097, 3.18816530161154160413273479350, 3.66254060472490055928832640638, 4.09648511318576568001315043004, 4.86788396244498416072826967746, 5.00401420807203998386668496800, 5.82865531445608805589229308253, 5.87631680476208146052773780967, 6.42877478120596479587860185234, 6.47840116003058930629506805179, 7.10033155884986463762739968618, 7.69637422811741651850934558385, 8.021975407540844688834874989070, 8.166221422442874923142527804299, 8.973240765902173635808131042840, 9.409849107021290924212913667598

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