Properties

Degree $2$
Conductor $33600$
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 − 7-s + 9-s − 4·11-s − 2·13-s − 2·17-s + 4·19-s + 21-s + 8·23-s − 27-s − 6·29-s − 8·31-s + 4·33-s − 2·37-s + 2·39-s + 2·41-s − 12·43-s + 8·47-s + 49-s + 2·51-s + 6·53-s − 4·57-s − 4·59-s + 2·61-s − 63-s + 12·67-s − 8·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.485·17-s + 0.917·19-s + 0.218·21-s + 1.66·23-s − 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.696·33-s − 0.328·37-s + 0.320·39-s + 0.312·41-s − 1.82·43-s + 1.16·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s − 0.529·57-s − 0.520·59-s + 0.256·61-s − 0.125·63-s + 1.46·67-s − 0.963·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33600\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{33600} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 33600,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 \)
7 \( 1 + T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
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

−15.22970623553413, −15.01385939584636, −14.13406280315785, −13.55923828069938, −13.02834120621349, −12.74732959858203, −12.14308079113538, −11.48372327899064, −10.93485193373410, −10.65294786486582, −9.933799654818406, −9.354821304934509, −9.017390297648957, −8.059940443153103, −7.629083395411812, −6.861993280901185, −6.766996285682923, −5.516779965800145, −5.388114981750604, −4.897282613421039, −3.927772768792639, −3.326096328712812, −2.580441807160393, −1.889820560747848, −0.8140051888799191, 0, 0.8140051888799191, 1.889820560747848, 2.580441807160393, 3.326096328712812, 3.927772768792639, 4.897282613421039, 5.388114981750604, 5.516779965800145, 6.766996285682923, 6.861993280901185, 7.629083395411812, 8.059940443153103, 9.017390297648957, 9.354821304934509, 9.933799654818406, 10.65294786486582, 10.93485193373410, 11.48372327899064, 12.14308079113538, 12.74732959858203, 13.02834120621349, 13.55923828069938, 14.13406280315785, 15.01385939584636, 15.22970623553413

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