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 + 6·13-s − 2·17-s − 4·19-s − 21-s + 8·23-s + 27-s + 2·29-s − 4·33-s − 10·37-s + 6·39-s − 6·41-s + 4·43-s + 49-s − 2·51-s + 6·53-s − 4·57-s + 4·59-s − 6·61-s − 63-s − 4·67-s + 8·69-s − 8·71-s − 10·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 1.66·13-s − 0.485·17-s − 0.917·19-s − 0.218·21-s + 1.66·23-s + 0.192·27-s + 0.371·29-s − 0.696·33-s − 1.64·37-s + 0.960·39-s − 0.937·41-s + 0.609·43-s + 1/7·49-s − 0.280·51-s + 0.824·53-s − 0.529·57-s + 0.520·59-s − 0.768·61-s − 0.125·63-s − 0.488·67-s + 0.963·69-s − 0.949·71-s − 1.17·73-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 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−15.34105036915970, −14.87768611911133, −14.08766471385324, −13.57641829089569, −13.13469904945172, −12.94507703884195, −12.21072167586817, −11.46521871152165, −10.87448627020241, −10.44633522957937, −10.12520659042612, −9.049411237826015, −8.784709845346452, −8.464907877941447, −7.663293919055153, −7.087716052136132, −6.530192601434237, −5.927454983175663, −5.216469105226557, −4.608652977451166, −3.846631629274388, −3.246437768248460, −2.712034310435650, −1.905385633567923, −1.087770642663804, 0, 1.087770642663804, 1.905385633567923, 2.712034310435650, 3.246437768248460, 3.846631629274388, 4.608652977451166, 5.216469105226557, 5.927454983175663, 6.530192601434237, 7.087716052136132, 7.663293919055153, 8.464907877941447, 8.784709845346452, 9.049411237826015, 10.12520659042612, 10.44633522957937, 10.87448627020241, 11.46521871152165, 12.21072167586817, 12.94507703884195, 13.13469904945172, 13.57641829089569, 14.08766471385324, 14.87768611911133, 15.34105036915970

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