Properties

Degree $2$
Conductor $216384$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·5-s + 9-s + 4·11-s − 2·13-s + 2·15-s + 6·17-s + 4·19-s − 23-s − 25-s − 27-s + 2·29-s + 8·31-s − 4·33-s − 6·37-s + 2·39-s + 6·41-s + 4·43-s − 2·45-s + 8·47-s − 6·51-s − 6·53-s − 8·55-s − 4·57-s + 4·59-s − 10·61-s + 4·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.516·15-s + 1.45·17-s + 0.917·19-s − 0.208·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.696·33-s − 0.986·37-s + 0.320·39-s + 0.937·41-s + 0.609·43-s − 0.298·45-s + 1.16·47-s − 0.840·51-s − 0.824·53-s − 1.07·55-s − 0.529·57-s + 0.520·59-s − 1.28·61-s + 0.496·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216384\)    =    \(2^{6} \cdot 3 \cdot 7^{2} \cdot 23\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{216384} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 216384,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.769607870\)
\(L(\frac12)\) \(\approx\) \(1.769607870\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
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

−12.66532062672228, −12.31239329275891, −12.05076281853130, −11.72846389607822, −11.28752896330903, −10.69752786244334, −10.14827009687788, −9.792642158991737, −9.274904320539725, −8.809736728586443, −8.034180621471110, −7.791079861040377, −7.273412547900058, −6.838091420316203, −6.240016585947941, −5.709076423198807, −5.320176677659584, −4.535314030579514, −4.244096031520385, −3.673393953343792, −3.113404606283938, −2.568762504285280, −1.508412906310101, −1.126157949994292, −0.4436669396692323, 0.4436669396692323, 1.126157949994292, 1.508412906310101, 2.568762504285280, 3.113404606283938, 3.673393953343792, 4.244096031520385, 4.535314030579514, 5.320176677659584, 5.709076423198807, 6.240016585947941, 6.838091420316203, 7.273412547900058, 7.791079861040377, 8.034180621471110, 8.809736728586443, 9.274904320539725, 9.792642158991737, 10.14827009687788, 10.69752786244334, 11.28752896330903, 11.72846389607822, 12.05076281853130, 12.31239329275891, 12.66532062672228

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