Properties

Label 4-1440e2-1.1-c1e2-0-23
Degree $4$
Conductor $2073600$
Sign $1$
Analytic cond. $132.214$
Root an. cond. $3.39093$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 8·13-s − 4·17-s + 3·25-s + 12·29-s + 16·37-s − 16·41-s + 6·49-s + 12·53-s + 20·61-s − 16·65-s + 12·73-s + 8·85-s − 8·89-s + 4·97-s − 20·101-s − 12·109-s − 28·113-s − 2·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s − 24·145-s + 149-s + 151-s + ⋯
L(s)  = 1  − 0.894·5-s + 2.21·13-s − 0.970·17-s + 3/5·25-s + 2.22·29-s + 2.63·37-s − 2.49·41-s + 6/7·49-s + 1.64·53-s + 2.56·61-s − 1.98·65-s + 1.40·73-s + 0.867·85-s − 0.847·89-s + 0.406·97-s − 1.99·101-s − 1.14·109-s − 2.63·113-s − 0.181·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.99·145-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2073600\)    =    \(2^{10} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(132.214\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2073600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.243947625\)
\(L(\frac12)\) \(\approx\) \(2.243947625\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
good7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.599542367778113449539114677714, −9.373791719654431270600819972011, −8.654771683914173419698468705002, −8.495905321502352456562376203856, −8.166052791193086010050666688408, −8.064085360155866759608363816890, −7.16226301259578703183065798768, −6.85326970006485740939466684428, −6.48775262574578907813008021114, −6.26337537079386845543709924020, −5.41073814013941971526674776398, −5.34944635106809400960610371164, −4.41054170800755004985245042305, −4.20678341357415009135671802046, −3.86396293560103417260816712209, −3.23572458576821090731290642935, −2.75396112564473314256842278454, −2.12132454984846043350366901398, −1.16402186534327768941641869551, −0.72225113761563103121873608148, 0.72225113761563103121873608148, 1.16402186534327768941641869551, 2.12132454984846043350366901398, 2.75396112564473314256842278454, 3.23572458576821090731290642935, 3.86396293560103417260816712209, 4.20678341357415009135671802046, 4.41054170800755004985245042305, 5.34944635106809400960610371164, 5.41073814013941971526674776398, 6.26337537079386845543709924020, 6.48775262574578907813008021114, 6.85326970006485740939466684428, 7.16226301259578703183065798768, 8.064085360155866759608363816890, 8.166052791193086010050666688408, 8.495905321502352456562376203856, 8.654771683914173419698468705002, 9.373791719654431270600819972011, 9.599542367778113449539114677714

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