Properties

Degree $2$
Conductor $235200$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s − 4·11-s + 2·13-s + 2·17-s − 4·19-s + 8·23-s + 27-s − 6·29-s + 8·31-s − 4·33-s − 2·37-s + 2·39-s − 2·41-s − 12·43-s − 8·47-s + 2·51-s + 6·53-s − 4·57-s + 4·59-s − 2·61-s + 12·67-s + 8·69-s + 8·71-s − 14·73-s + 81-s − 12·83-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.485·17-s − 0.917·19-s + 1.66·23-s + 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.696·33-s − 0.328·37-s + 0.320·39-s − 0.312·41-s − 1.82·43-s − 1.16·47-s + 0.280·51-s + 0.824·53-s − 0.529·57-s + 0.520·59-s − 0.256·61-s + 1.46·67-s + 0.963·69-s + 0.949·71-s − 1.63·73-s + 1/9·81-s − 1.31·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(235200\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{235200} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 235200,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.22102116909771, −12.84704361039712, −12.38178307681530, −11.52025988487140, −11.44887489748148, −10.66508678592027, −10.40639105075954, −9.862815801178429, −9.492844333478485, −8.712077946780768, −8.458053444689030, −8.126971706948199, −7.515998193943051, −6.939319796371834, −6.659769467124306, −5.904590181989305, −5.409465258350829, −4.869104713016899, −4.497875677085325, −3.649669383374439, −3.287167153393678, −2.766863831708448, −2.162810724085006, −1.560783335831076, −0.8343075913586646, 0, 0.8343075913586646, 1.560783335831076, 2.162810724085006, 2.766863831708448, 3.287167153393678, 3.649669383374439, 4.497875677085325, 4.869104713016899, 5.409465258350829, 5.904590181989305, 6.659769467124306, 6.939319796371834, 7.515998193943051, 8.126971706948199, 8.458053444689030, 8.712077946780768, 9.492844333478485, 9.862815801178429, 10.40639105075954, 10.66508678592027, 11.44887489748148, 11.52025988487140, 12.38178307681530, 12.84704361039712, 13.22102116909771

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