Properties

Label 4-1560e2-1.1-c0e2-0-1
Degree $4$
Conductor $2433600$
Sign $1$
Analytic cond. $0.606126$
Root an. cond. $0.882349$
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-s − 3-s − 2·5-s − 6-s − 8-s − 2·10-s − 11-s + 2·13-s + 2·15-s − 16-s + 2·17-s − 22-s − 23-s + 24-s + 3·25-s + 2·26-s + 27-s − 29-s + 2·30-s − 2·31-s + 33-s + 2·34-s + 37-s − 2·39-s + 2·40-s + 43-s − 46-s + ⋯
L(s)  = 1  + 2-s − 3-s − 2·5-s − 6-s − 8-s − 2·10-s − 11-s + 2·13-s + 2·15-s − 16-s + 2·17-s − 22-s − 23-s + 24-s + 3·25-s + 2·26-s + 27-s − 29-s + 2·30-s − 2·31-s + 33-s + 2·34-s + 37-s − 2·39-s + 2·40-s + 43-s − 46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2433600\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(0.606126\)
Root analytic conductor: \(0.882349\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2433600,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5714698676\)
\(L(\frac12)\) \(\approx\) \(0.5714698676\)
\(L(1)\) 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$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 + T + T^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
13$C_1$ \( ( 1 - T )^{2} \)
good7$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
11$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
17$C_2$ \( ( 1 - T + T^{2} )^{2} \)
19$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
23$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
29$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
31$C_2$ \( ( 1 + T + T^{2} )^{2} \)
37$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
41$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
43$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
47$C_2$ \( ( 1 - T + T^{2} )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
61$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
67$C_2$ \( ( 1 + T + T^{2} )^{2} \)
71$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_2$ \( ( 1 + T + T^{2} )^{2} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
97$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
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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

−9.974784129365682638359242953395, −9.168553505774915029795332745985, −9.123490154657383279849162407230, −8.352971208711085777644249772539, −8.344047369398747222890570315653, −7.72631677512436013508027944563, −7.36272270622419921711122879124, −7.22218939015665074314621385008, −6.25395447626052788118103128782, −6.02318342050676670170937782801, −5.59166644393434739119604911868, −5.52104663999998049994575977915, −4.75654329776446243524431860871, −4.40237697517048307052104126073, −3.86577231823986330470112004461, −3.63596889452085221097128607364, −3.16942435994920246102566525812, −2.74892321565279398541930181665, −1.47940075909707004064507090842, −0.58333095216195354247331909647, 0.58333095216195354247331909647, 1.47940075909707004064507090842, 2.74892321565279398541930181665, 3.16942435994920246102566525812, 3.63596889452085221097128607364, 3.86577231823986330470112004461, 4.40237697517048307052104126073, 4.75654329776446243524431860871, 5.52104663999998049994575977915, 5.59166644393434739119604911868, 6.02318342050676670170937782801, 6.25395447626052788118103128782, 7.22218939015665074314621385008, 7.36272270622419921711122879124, 7.72631677512436013508027944563, 8.344047369398747222890570315653, 8.352971208711085777644249772539, 9.123490154657383279849162407230, 9.168553505774915029795332745985, 9.974784129365682638359242953395

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