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Properties

Label 4-800e2-1.1-c0e2-0-3
Degree $4$
Conductor $640000$
Sign $1$
Analytic cond. $0.159402$
Root an. cond. $0.631863$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(640000\)    =    \(2^{10} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(0.159402\)
Root analytic conductor: \(0.631863\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 640000,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.036257166\)
\(L(\frac12)\) \(\approx\) \(1.036257166\)
\(L(1)\) 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

−10.59992151531658934819817186097, −10.37508677141032781256638495775, −9.822517529218652472810256623789, −9.473674477475567967323353023605, −8.917100478281945147692390632369, −8.572212431423190890905315262018, −8.077543216265555129976011944201, −7.902183045738170981526358898506, −7.17792146549312341326590544182, −6.85352309348448355130523764019, −6.20513009170500104962279076153, −5.92165793282480784092446772764, −5.35050401038494667986764952731, −5.09689581356146083295485265421, −4.06483515617659980911558398079, −3.94475876647355079220750533822, −3.05444402249148542367272708883, −3.04004687598663522254644475155, −1.58849934508451499306883648860, −1.36895002068491677992741418241, 1.36895002068491677992741418241, 1.58849934508451499306883648860, 3.04004687598663522254644475155, 3.05444402249148542367272708883, 3.94475876647355079220750533822, 4.06483515617659980911558398079, 5.09689581356146083295485265421, 5.35050401038494667986764952731, 5.92165793282480784092446772764, 6.20513009170500104962279076153, 6.85352309348448355130523764019, 7.17792146549312341326590544182, 7.902183045738170981526358898506, 8.077543216265555129976011944201, 8.572212431423190890905315262018, 8.917100478281945147692390632369, 9.473674477475567967323353023605, 9.822517529218652472810256623789, 10.37508677141032781256638495775, 10.59992151531658934819817186097

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