Properties

Label 4-532400-1.1-c1e2-0-9
Degree $4$
Conductor $532400$
Sign $-1$
Analytic cond. $33.9462$
Root an. cond. $2.41378$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 2·4-s + 5-s + 4·6-s − 3·9-s − 2·10-s + 11-s − 4·12-s + 8·13-s − 2·15-s − 4·16-s − 4·17-s + 6·18-s + 2·20-s − 2·22-s − 2·23-s − 4·25-s − 16·26-s + 14·27-s + 4·30-s + 8·32-s − 2·33-s + 8·34-s − 6·36-s − 16·39-s + 2·44-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 4-s + 0.447·5-s + 1.63·6-s − 9-s − 0.632·10-s + 0.301·11-s − 1.15·12-s + 2.21·13-s − 0.516·15-s − 16-s − 0.970·17-s + 1.41·18-s + 0.447·20-s − 0.426·22-s − 0.417·23-s − 4/5·25-s − 3.13·26-s + 2.69·27-s + 0.730·30-s + 1.41·32-s − 0.348·33-s + 1.37·34-s − 36-s − 2.56·39-s + 0.301·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(532400\)    =    \(2^{4} \cdot 5^{2} \cdot 11^{3}\)
Sign: $-1$
Analytic conductor: \(33.9462\)
Root analytic conductor: \(2.41378\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 532400,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T + p T^{2} \)
5$C_2$ \( 1 - T + p T^{2} \)
11$C_1$ \( 1 - T \)
good3$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
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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

−8.593732589964329387984594611773, −7.895851651484576537227893446235, −7.52743412041385429223075067799, −6.68528518558436973314151287167, −6.36261389471308870138602900888, −6.19560577468643576321422166894, −5.60369423645831790828591066554, −5.27601274318804963861581294420, −4.40338308258317956714309844682, −3.99802172535608567716568957865, −3.17341176038083221464044877547, −2.45229307003030663107907497428, −1.67179714482792995515547904518, −0.947006685949828438104025689659, 0, 0.947006685949828438104025689659, 1.67179714482792995515547904518, 2.45229307003030663107907497428, 3.17341176038083221464044877547, 3.99802172535608567716568957865, 4.40338308258317956714309844682, 5.27601274318804963861581294420, 5.60369423645831790828591066554, 6.19560577468643576321422166894, 6.36261389471308870138602900888, 6.68528518558436973314151287167, 7.52743412041385429223075067799, 7.895851651484576537227893446235, 8.593732589964329387984594611773

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