Properties

Label 4-40e4-1.1-c1e2-0-13
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 $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s − 4·7-s + 8·9-s − 2·13-s + 10·17-s − 8·19-s − 16·21-s − 4·23-s + 12·27-s + 2·37-s − 8·39-s − 12·43-s + 4·47-s + 8·49-s + 40·51-s − 14·53-s − 32·57-s − 8·59-s + 8·61-s − 32·63-s − 20·67-s − 16·69-s + 6·73-s + 32·79-s + 23·81-s − 4·83-s + 8·91-s + ⋯
L(s)  = 1  + 2.30·3-s − 1.51·7-s + 8/3·9-s − 0.554·13-s + 2.42·17-s − 1.83·19-s − 3.49·21-s − 0.834·23-s + 2.30·27-s + 0.328·37-s − 1.28·39-s − 1.82·43-s + 0.583·47-s + 8/7·49-s + 5.60·51-s − 1.92·53-s − 4.23·57-s − 1.04·59-s + 1.02·61-s − 4.03·63-s − 2.44·67-s − 1.92·69-s + 0.702·73-s + 3.60·79-s + 23/9·81-s − 0.439·83-s + 0.838·91-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: induced by $\chi_{1600} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2560000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.742694681\)
\(L(\frac12)\) \(\approx\) \(3.742694681\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
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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

−9.760988051984675591341229533649, −9.056634582381015403254671607378, −8.890910436929955606763598500602, −8.477534479365718684976487729233, −7.977162426659782429432604116678, −7.66347603819499225281650247792, −7.62156143975848594067059768836, −6.82835382930067116706614921599, −6.39502234215916413787621248104, −6.18644732074603755814908934176, −5.59966231196990228066846491567, −4.86483868430625068173221651437, −4.50453359516864159349933091629, −3.71583613784677420748047545493, −3.55413398289087669965633427385, −2.99841847909133225820363591973, −2.97129237650366007108472251230, −2.01273877255625461573063174726, −1.88358072592583766918140647479, −0.61623069282738966788727087392, 0.61623069282738966788727087392, 1.88358072592583766918140647479, 2.01273877255625461573063174726, 2.97129237650366007108472251230, 2.99841847909133225820363591973, 3.55413398289087669965633427385, 3.71583613784677420748047545493, 4.50453359516864159349933091629, 4.86483868430625068173221651437, 5.59966231196990228066846491567, 6.18644732074603755814908934176, 6.39502234215916413787621248104, 6.82835382930067116706614921599, 7.62156143975848594067059768836, 7.66347603819499225281650247792, 7.977162426659782429432604116678, 8.477534479365718684976487729233, 8.890910436929955606763598500602, 9.056634582381015403254671607378, 9.760988051984675591341229533649

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