Properties

Label 4-40e4-1.1-c1e2-0-19
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  − 2·3-s − 6·7-s + 2·9-s + 12·21-s + 2·23-s − 6·27-s + 24·41-s + 18·43-s + 14·47-s + 18·49-s + 16·61-s − 12·63-s + 6·67-s − 4·69-s + 11·81-s − 22·83-s + 36·101-s − 18·103-s + 26·107-s + 22·121-s − 48·123-s + 127-s − 36·129-s + 131-s + 137-s + 139-s − 28·141-s + ⋯
L(s)  = 1  − 1.15·3-s − 2.26·7-s + 2/3·9-s + 2.61·21-s + 0.417·23-s − 1.15·27-s + 3.74·41-s + 2.74·43-s + 2.04·47-s + 18/7·49-s + 2.04·61-s − 1.51·63-s + 0.733·67-s − 0.481·69-s + 11/9·81-s − 2.41·83-s + 3.58·101-s − 1.77·103-s + 2.51·107-s + 2·121-s − 4.32·123-s + 0.0887·127-s − 3.16·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.35·141-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: \(0\)
Selberg data: \((4,\ 2560000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.079918778\)
\(L(\frac12)\) \(\approx\) \(1.079918778\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2^2$ \( 1 + p^{2} T^{4} \)
17$C_2^2$ \( 1 + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - p T^{2} )^{2} \)
37$C_2^2$ \( 1 + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2^2$ \( 1 + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + p^{2} T^{4} \)
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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

−9.604994132106875945435616786468, −9.378468980958517742437818312667, −8.908489870436310052046291335253, −8.676683686322070143775019249456, −7.67013848081013162138557875546, −7.58391356396562637697442744786, −7.09655167360036780894579923693, −6.79580121814763522725822710827, −6.18119996349557034845715173650, −5.92255289785252634124047291822, −5.73648311835224548715439573185, −5.38943781518919203046266981553, −4.35635188458020468337971909209, −4.29392024536477669311303045219, −3.74131552264969916001740768214, −3.17512173195372653193629663407, −2.56100977990067252553594394880, −2.24603293672649670424589853110, −0.78258933533769630609459500687, −0.66370069951966065907765649761, 0.66370069951966065907765649761, 0.78258933533769630609459500687, 2.24603293672649670424589853110, 2.56100977990067252553594394880, 3.17512173195372653193629663407, 3.74131552264969916001740768214, 4.29392024536477669311303045219, 4.35635188458020468337971909209, 5.38943781518919203046266981553, 5.73648311835224548715439573185, 5.92255289785252634124047291822, 6.18119996349557034845715173650, 6.79580121814763522725822710827, 7.09655167360036780894579923693, 7.58391356396562637697442744786, 7.67013848081013162138557875546, 8.676683686322070143775019249456, 8.908489870436310052046291335253, 9.378468980958517742437818312667, 9.604994132106875945435616786468

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