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 − 6·11-s + 6·17-s − 4·19-s + 6·23-s − 27-s + 2·29-s − 8·31-s + 6·33-s − 2·37-s − 10·41-s + 12·43-s − 8·47-s − 6·51-s − 2·53-s + 4·57-s − 4·59-s − 8·61-s + 16·67-s − 6·69-s + 10·71-s − 4·79-s + 81-s − 4·83-s − 2·87-s + 6·89-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.80·11-s + 1.45·17-s − 0.917·19-s + 1.25·23-s − 0.192·27-s + 0.371·29-s − 1.43·31-s + 1.04·33-s − 0.328·37-s − 1.56·41-s + 1.82·43-s − 1.16·47-s − 0.840·51-s − 0.274·53-s + 0.529·57-s − 0.520·59-s − 1.02·61-s + 1.95·67-s − 0.722·69-s + 1.18·71-s − 0.450·79-s + 1/9·81-s − 0.439·83-s − 0.214·87-s + 0.635·89-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 \( 1 + 6 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−12.86853192740105, −12.75144389374165, −12.38765565179813, −11.74702833789763, −11.10680047956838, −10.84883339723318, −10.46669981016190, −9.976471638007014, −9.543857167137447, −8.915625872095284, −8.370295206842269, −7.859411696404859, −7.558009433503510, −6.945248485589409, −6.517627451445118, −5.768266153191413, −5.449157627305181, −5.019632008035785, −4.623395318783201, −3.775001022549818, −3.280555858142379, −2.750885945647112, −2.095522271504491, −1.443930999387790, −0.6570283558737174, 0, 0.6570283558737174, 1.443930999387790, 2.095522271504491, 2.750885945647112, 3.280555858142379, 3.775001022549818, 4.623395318783201, 5.019632008035785, 5.449157627305181, 5.768266153191413, 6.517627451445118, 6.945248485589409, 7.558009433503510, 7.859411696404859, 8.370295206842269, 8.915625872095284, 9.543857167137447, 9.976471638007014, 10.46669981016190, 10.84883339723318, 11.10680047956838, 11.74702833789763, 12.38765565179813, 12.75144389374165, 12.86853192740105

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