Properties

Degree $4$
Conductor $9144576$
Sign $1$
Motivic weight $0$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 3·19-s − 25-s + 3·31-s + 2·37-s − 2·43-s − 3·61-s + 2·67-s − 3·73-s + 2·79-s − 109-s + 121-s + 127-s + 131-s + 3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 175-s + 179-s + 181-s + ⋯
L(s)  = 1  − 7-s − 3·19-s − 25-s + 3·31-s + 2·37-s − 2·43-s − 3·61-s + 2·67-s − 3·73-s + 2·79-s − 109-s + 121-s + 127-s + 131-s + 3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 175-s + 179-s + 181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9144576\)    =    \(2^{8} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(0\)
Character: induced by $\chi_{3024} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9144576,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7895972654\)
\(L(\frac12)\) \(\approx\) \(0.7895972654\)
\(L(1)\) 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 \( 1 \)
7$C_2$ \( 1 + T + T^{2} \)
good5$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
11$C_2^2$ \( 1 - T^{2} + T^{4} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
19$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T + T^{2} ) \)
23$C_2^2$ \( 1 - T^{2} + T^{4} \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 - T + T^{2} ) \)
37$C_2$ \( ( 1 - T + T^{2} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2$ \( ( 1 + T + T^{2} )^{2} \)
47$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
53$C_2^2$ \( 1 - T^{2} + T^{4} \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T + T^{2} ) \)
67$C_2$ \( ( 1 - T + T^{2} )^{2} \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T + T^{2} ) \)
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

−9.054177946641286646910082250703, −8.648945698951114820520293425504, −8.362177901967644800640795167687, −7.965417739014474246010891126742, −7.80051716747258995739023934599, −7.15165673939126676052198142380, −6.49293798176864093981901029232, −6.44975179771677594213385084891, −6.33048902036013736782946845390, −5.85022465390939629681901195272, −5.29546437456800243625744192685, −4.49439022163802085464188620348, −4.48827454076173295228232081585, −4.20535541113172442382398115999, −3.45997084599980878527322066409, −3.02922798616085085357130631764, −2.63365814456375771932375669693, −2.09677632069388811530758308037, −1.57661444517640536435648763308, −0.53659147058404712339798999000, 0.53659147058404712339798999000, 1.57661444517640536435648763308, 2.09677632069388811530758308037, 2.63365814456375771932375669693, 3.02922798616085085357130631764, 3.45997084599980878527322066409, 4.20535541113172442382398115999, 4.48827454076173295228232081585, 4.49439022163802085464188620348, 5.29546437456800243625744192685, 5.85022465390939629681901195272, 6.33048902036013736782946845390, 6.44975179771677594213385084891, 6.49293798176864093981901029232, 7.15165673939126676052198142380, 7.80051716747258995739023934599, 7.965417739014474246010891126742, 8.362177901967644800640795167687, 8.648945698951114820520293425504, 9.054177946641286646910082250703

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