Properties

Degree $2$
Conductor $216384$
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 − 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: \(1\)
Selberg data: \((2,\ 216384,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
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

−13.09582682894599, −12.71358711818493, −12.33188518074709, −12.02630842490542, −11.23537626942775, −10.90141583396037, −10.46011994784565, −9.858840123489355, −9.579271739844344, −8.897701293041354, −8.358957202089758, −7.983960324239386, −7.559668205481556, −7.336384579756811, −6.576732147417778, −6.031492985203461, −5.276093278762327, −5.015868547175418, −4.389622893252406, −3.746684374737474, −3.292277978880189, −2.906150695845250, −2.099456955283116, −1.674376091227832, −0.6412142100979749, 0, 0.6412142100979749, 1.674376091227832, 2.099456955283116, 2.906150695845250, 3.292277978880189, 3.746684374737474, 4.389622893252406, 5.015868547175418, 5.276093278762327, 6.031492985203461, 6.576732147417778, 7.336384579756811, 7.559668205481556, 7.983960324239386, 8.358957202089758, 8.897701293041354, 9.579271739844344, 9.858840123489355, 10.46011994784565, 10.90141583396037, 11.23537626942775, 12.02630842490542, 12.33188518074709, 12.71358711818493, 13.09582682894599

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