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

−15.19942093159856, −14.75650853400172, −14.27926641368770, −13.84600192837762, −13.24765435570506, −12.98952079571088, −12.20638978688179, −11.60522342944015, −11.05174132441280, −10.53258632923169, −9.952004817943698, −9.231621618985570, −8.925866546574860, −8.520071459489707, −7.786957293185510, −6.994644563085540, −6.483970292759363, −6.122375049325342, −5.271486820400391, −4.587300844243689, −3.959674969308091, −3.431933376402704, −2.413992834565775, −1.827125722142128, −1.422095868063140, 0, 1.422095868063140, 1.827125722142128, 2.413992834565775, 3.431933376402704, 3.959674969308091, 4.587300844243689, 5.271486820400391, 6.122375049325342, 6.483970292759363, 6.994644563085540, 7.786957293185510, 8.520071459489707, 8.925866546574860, 9.231621618985570, 9.952004817943698, 10.53258632923169, 11.05174132441280, 11.60522342944015, 12.20638978688179, 12.98952079571088, 13.24765435570506, 13.84600192837762, 14.27926641368770, 14.75650853400172, 15.19942093159856

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