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Properties

Label 4-288e2-1.1-c0e2-0-1
Degree $4$
Conductor $82944$
Sign $1$
Analytic cond. $0.0206585$
Root an. cond. $0.379118$
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  + 3-s − 11-s − 2·17-s + 2·19-s − 25-s − 27-s − 33-s + 41-s − 43-s − 49-s − 2·51-s + 2·57-s − 59-s − 67-s − 2·73-s − 75-s − 81-s + 2·83-s + 4·89-s + 97-s + 2·107-s − 2·113-s + 121-s + 123-s + 127-s − 129-s + 131-s + ⋯
L(s)  = 1  + 3-s − 11-s − 2·17-s + 2·19-s − 25-s − 27-s − 33-s + 41-s − 43-s − 49-s − 2·51-s + 2·57-s − 59-s − 67-s − 2·73-s − 75-s − 81-s + 2·83-s + 4·89-s + 97-s + 2·107-s − 2·113-s + 121-s + 123-s + 127-s − 129-s + 131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(82944\)    =    \(2^{10} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(0.0206585\)
Root analytic conductor: \(0.379118\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 82944,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7211143485\)
\(L(\frac12)\) \(\approx\) \(0.7211143485\)
\(L(1)\) 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 \( 1 \)
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

−12.01611523951067512914040256204, −11.97573815334376466413570916088, −11.20665783989842032015504872554, −11.01712646271616680217059782019, −10.30123294405171983945319577700, −9.868532226469307474670523599663, −9.265452115718542584533803237753, −9.063727139643326893947662893369, −8.492488601413409686406975514725, −7.917735007051806506900779299386, −7.52059976324434673095561067634, −7.22493652639742857624836174268, −6.17276682240831828685795082405, −6.02084406712930474359488698716, −4.94752894790484327388806822254, −4.80648808073003878039043101676, −3.76292847751230084723448605318, −3.23161892249714255438115202390, −2.55740197511479142388327182984, −1.88464288874454619446146495074, 1.88464288874454619446146495074, 2.55740197511479142388327182984, 3.23161892249714255438115202390, 3.76292847751230084723448605318, 4.80648808073003878039043101676, 4.94752894790484327388806822254, 6.02084406712930474359488698716, 6.17276682240831828685795082405, 7.22493652639742857624836174268, 7.52059976324434673095561067634, 7.917735007051806506900779299386, 8.492488601413409686406975514725, 9.063727139643326893947662893369, 9.265452115718542584533803237753, 9.868532226469307474670523599663, 10.30123294405171983945319577700, 11.01712646271616680217059782019, 11.20665783989842032015504872554, 11.97573815334376466413570916088, 12.01611523951067512914040256204

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