Properties

Degree $4$
Conductor $451584$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5·7-s + 2·11-s + 2·13-s + 2·17-s − 5·19-s − 5·21-s − 6·23-s + 5·25-s − 27-s − 16·29-s + 3·31-s + 2·33-s + 9·37-s + 2·39-s + 4·41-s + 2·43-s − 8·47-s + 18·49-s + 2·51-s − 6·53-s − 5·57-s − 6·59-s + 2·61-s + 5·67-s − 6·69-s + 8·71-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.88·7-s + 0.603·11-s + 0.554·13-s + 0.485·17-s − 1.14·19-s − 1.09·21-s − 1.25·23-s + 25-s − 0.192·27-s − 2.97·29-s + 0.538·31-s + 0.348·33-s + 1.47·37-s + 0.320·39-s + 0.624·41-s + 0.304·43-s − 1.16·47-s + 18/7·49-s + 0.280·51-s − 0.824·53-s − 0.662·57-s − 0.781·59-s + 0.256·61-s + 0.610·67-s − 0.722·69-s + 0.949·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(451584\)    =    \(2^{10} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{672} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 451584,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44558\)
\(L(\frac12)\) \(\approx\) \(1.44558\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 + 5 T + p T^{2} \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 3 T - 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 9 T + 44 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 5 T - 54 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
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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

−10.67769103457568103418758291920, −10.18987665732156992179273476391, −9.660380412515403207056694063490, −9.355376221145531575895543091884, −9.289119926915806724691452792594, −8.547433564021256449521646652138, −8.245822510331323119501921809955, −7.57842880539696012350789495096, −7.32630130891329094292412796946, −6.58152171388148417110033479455, −6.20206266565479074778989239588, −6.08374990537256173628167383456, −5.44262333822350199102950112321, −4.57356004989035895150065611371, −4.03603809203216524183816860156, −3.45402249761594588939019684739, −3.35911810172884238495230737964, −2.41844532142829202541071407104, −1.89900277113488870219400588468, −0.60812738537787409802359400123, 0.60812738537787409802359400123, 1.89900277113488870219400588468, 2.41844532142829202541071407104, 3.35911810172884238495230737964, 3.45402249761594588939019684739, 4.03603809203216524183816860156, 4.57356004989035895150065611371, 5.44262333822350199102950112321, 6.08374990537256173628167383456, 6.20206266565479074778989239588, 6.58152171388148417110033479455, 7.32630130891329094292412796946, 7.57842880539696012350789495096, 8.245822510331323119501921809955, 8.547433564021256449521646652138, 9.289119926915806724691452792594, 9.355376221145531575895543091884, 9.660380412515403207056694063490, 10.18987665732156992179273476391, 10.67769103457568103418758291920

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