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Properties

Label 4-1280e2-1.1-c0e2-0-5
Degree $4$
Conductor $1638400$
Sign $1$
Analytic cond. $0.408069$
Root an. cond. $0.799251$
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  + 2·5-s − 2·13-s + 2·17-s + 3·25-s − 2·37-s − 2·53-s + 4·61-s − 4·65-s + 2·73-s − 81-s + 4·85-s − 2·97-s − 2·113-s − 2·121-s + 4·125-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 + ⋯
L(s)  = 1  + 2·5-s − 2·13-s + 2·17-s + 3·25-s − 2·37-s − 2·53-s + 4·61-s − 4·65-s + 2·73-s − 81-s + 4·85-s − 2·97-s − 2·113-s − 2·121-s + 4·125-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1638400\)    =    \(2^{16} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(0.408069\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1638400,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.524494646\)
\(L(\frac12)\) \(\approx\) \(1.524494646\)
\(L(1)\) 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$C_1$ \( ( 1 - T )^{2} \)
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$C_2$ \( ( 1 + T^{2} )^{2} \)
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$C_1$ \( ( 1 - T )^{4} \)
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

−9.964832605681071972461891305552, −9.799743619951564699777668367968, −9.291224714006104767552104887300, −9.186141973908284260333454055670, −8.291711220449639671757808819552, −8.248164377289671937300698850356, −7.52536394127897905866697190257, −7.17438654156441139716822791752, −6.59235801628521209496711360420, −6.57314167631080201188453586388, −5.70376606193782252549884416415, −5.41866129202010967020773057736, −5.05250576282773688095770727869, −4.99125058016544368812370535309, −3.96212254881160499698511916202, −3.43923422826357538745900098234, −2.70701320733826131018143723953, −2.48842071831408910394253778505, −1.77841542129944846033303856091, −1.20109791377981409238931281914, 1.20109791377981409238931281914, 1.77841542129944846033303856091, 2.48842071831408910394253778505, 2.70701320733826131018143723953, 3.43923422826357538745900098234, 3.96212254881160499698511916202, 4.99125058016544368812370535309, 5.05250576282773688095770727869, 5.41866129202010967020773057736, 5.70376606193782252549884416415, 6.57314167631080201188453586388, 6.59235801628521209496711360420, 7.17438654156441139716822791752, 7.52536394127897905866697190257, 8.248164377289671937300698850356, 8.291711220449639671757808819552, 9.186141973908284260333454055670, 9.291224714006104767552104887300, 9.799743619951564699777668367968, 9.964832605681071972461891305552

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