Properties

Label 4-72e2-1.1-c0e2-0-0
Degree $4$
Conductor $5184$
Sign $1$
Analytic cond. $0.00129115$
Root an. cond. $0.189559$
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 + 6-s + 8-s + 11-s − 16-s − 2·17-s − 2·19-s − 22-s − 24-s − 25-s + 27-s − 33-s + 2·34-s + 2·38-s + 41-s + 43-s + 48-s − 49-s + 50-s + 2·51-s − 54-s + 2·57-s + 59-s + 64-s + 66-s + 67-s + ⋯
L(s)  = 1  − 2-s − 3-s + 6-s + 8-s + 11-s − 16-s − 2·17-s − 2·19-s − 22-s − 24-s − 25-s + 27-s − 33-s + 2·34-s + 2·38-s + 41-s + 43-s + 48-s − 49-s + 50-s + 2·51-s − 54-s + 2·57-s + 59-s + 64-s + 66-s + 67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(0.00129115\)
Root analytic conductor: \(0.189559\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{72} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5184,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1256607866\)
\(L(\frac12)\) \(\approx\) \(0.1256607866\)
\(L(1)\) 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$C_2$ \( 1 + T + T^{2} \)
good5$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
7$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
11$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
13$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
17$C_2$ \( ( 1 + T + T^{2} )^{2} \)
19$C_2$ \( ( 1 + T + T^{2} )^{2} \)
23$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
29$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
31$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
43$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
47$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
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_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2$ \( ( 1 + T + T^{2} )^{2} \)
79$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
83$C_2$ \( ( 1 + T + T^{2} )^{2} \)
89$C_1$ \( ( 1 - T )^{4} \)
97$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
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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

−15.34043328614189626901675784065, −14.49356786122127795921995434996, −14.34571041592205361528943903242, −13.25301324941013762299050072315, −13.14488498825266391224630030241, −12.39764694798549826442099294177, −11.46457120346775008809481426109, −11.43543784024726873689769103533, −10.58013683892231668704707658551, −10.42530837244236248432447086564, −9.352510038710622356505889286642, −9.026004063438331298527805431678, −8.479371373944274566855562461374, −7.78292414143922961147563368342, −6.70640051462313961974619854325, −6.52229259023448238669588328170, −5.63538510576822836356840705680, −4.47388984368426759808121563855, −4.17397350823654265270160808926, −2.13460287732343082794920450963, 2.13460287732343082794920450963, 4.17397350823654265270160808926, 4.47388984368426759808121563855, 5.63538510576822836356840705680, 6.52229259023448238669588328170, 6.70640051462313961974619854325, 7.78292414143922961147563368342, 8.479371373944274566855562461374, 9.026004063438331298527805431678, 9.352510038710622356505889286642, 10.42530837244236248432447086564, 10.58013683892231668704707658551, 11.43543784024726873689769103533, 11.46457120346775008809481426109, 12.39764694798549826442099294177, 13.14488498825266391224630030241, 13.25301324941013762299050072315, 14.34571041592205361528943903242, 14.49356786122127795921995434996, 15.34043328614189626901675784065

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