Properties

Degree $2$
Conductor $30912$
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 − 7-s + 9-s − 4·11-s + 2·13-s − 2·15-s − 6·17-s + 4·19-s + 21-s + 23-s − 25-s − 27-s + 2·29-s + 8·31-s + 4·33-s − 2·35-s − 6·37-s − 2·39-s − 6·41-s − 4·43-s + 2·45-s + 8·47-s + 49-s + 6·51-s − 6·53-s − 8·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s − 0.377·7-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 + 0.218·21-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 − 0.338·35-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 + 1/7·49-s + 0.840·51-s − 0.824·53-s − 1.07·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(30912\)    =    \(2^{6} \cdot 3 \cdot 7 \cdot 23\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{30912} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 30912,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
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

−15.51488372963165, −15.04254790988320, −13.90307886349619, −13.80525256539370, −13.29212580274043, −12.83058794746369, −12.23123110991322, −11.53114982054522, −11.13080314108051, −10.43901340667491, −10.03885322377946, −9.672219887256609, −8.785198622154931, −8.454783005975069, −7.644635717861100, −6.920208246854855, −6.507915866615559, −5.911330512676735, −5.296494437975712, −4.901057945466802, −4.097379959984436, −3.243305531865748, −2.538101165401875, −1.914522607705434, −0.9653886780361722, 0, 0.9653886780361722, 1.914522607705434, 2.538101165401875, 3.243305531865748, 4.097379959984436, 4.901057945466802, 5.296494437975712, 5.911330512676735, 6.507915866615559, 6.920208246854855, 7.644635717861100, 8.454783005975069, 8.785198622154931, 9.672219887256609, 10.03885322377946, 10.43901340667491, 11.13080314108051, 11.53114982054522, 12.23123110991322, 12.83058794746369, 13.29212580274043, 13.80525256539370, 13.90307886349619, 15.04254790988320, 15.51488372963165

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