Properties

Degree $2$
Conductor $162288$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 4·11-s + 2·13-s − 6·17-s + 4·19-s − 23-s − 25-s + 2·29-s − 8·31-s + 6·37-s − 6·41-s + 4·43-s + 8·47-s − 6·53-s + 8·55-s − 4·59-s + 10·61-s − 4·65-s − 4·67-s − 8·71-s + 6·73-s + 12·83-s + 12·85-s + 2·89-s − 8·95-s − 10·97-s + 101-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.20·11-s + 0.554·13-s − 1.45·17-s + 0.917·19-s − 0.208·23-s − 1/5·25-s + 0.371·29-s − 1.43·31-s + 0.986·37-s − 0.937·41-s + 0.609·43-s + 1.16·47-s − 0.824·53-s + 1.07·55-s − 0.520·59-s + 1.28·61-s − 0.496·65-s − 0.488·67-s − 0.949·71-s + 0.702·73-s + 1.31·83-s + 1.30·85-s + 0.211·89-s − 0.820·95-s − 1.01·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162288\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 23\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{162288} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 162288,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7224393877\)
\(L(\frac12)\) \(\approx\) \(0.7224393877\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
23 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
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

−13.32321241314529, −12.75982280859506, −12.39017000696464, −11.71866567181034, −11.37400309120713, −10.91261396381468, −10.59471382684710, −9.934549482233103, −9.416891141302010, −8.821594675597016, −8.485942811949406, −7.841880342667797, −7.491692252555096, −7.138978473095953, −6.337888291755544, −5.948761549800117, −5.246781413745522, −4.833979547473779, −4.203898668011491, −3.719700389047904, −3.186021892880348, −2.477938169184385, −2.002955240823668, −1.074520812528599, −0.2704051508450814, 0.2704051508450814, 1.074520812528599, 2.002955240823668, 2.477938169184385, 3.186021892880348, 3.719700389047904, 4.203898668011491, 4.833979547473779, 5.246781413745522, 5.948761549800117, 6.337888291755544, 7.138978473095953, 7.491692252555096, 7.841880342667797, 8.485942811949406, 8.821594675597016, 9.416891141302010, 9.934549482233103, 10.59471382684710, 10.91261396381468, 11.37400309120713, 11.71866567181034, 12.39017000696464, 12.75982280859506, 13.32321241314529

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