Properties

Degree $2$
Conductor $91200$
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 − 4·7-s + 9-s + 4·11-s − 2·13-s + 2·17-s + 19-s − 4·21-s + 8·23-s + 27-s − 6·29-s + 4·31-s + 4·33-s − 10·37-s − 2·39-s − 2·41-s + 12·43-s + 9·49-s + 2·51-s + 6·53-s + 57-s + 10·61-s − 4·63-s − 4·67-s + 8·69-s − 8·71-s − 2·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.485·17-s + 0.229·19-s − 0.872·21-s + 1.66·23-s + 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.696·33-s − 1.64·37-s − 0.320·39-s − 0.312·41-s + 1.82·43-s + 9/7·49-s + 0.280·51-s + 0.824·53-s + 0.132·57-s + 1.28·61-s − 0.503·63-s − 0.488·67-s + 0.963·69-s − 0.949·71-s − 0.234·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91200\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 19\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{91200} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 91200,\ (\ :1/2),\ -1)\)

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 \( 1 \)
19 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−14.07330232599005, −13.57769385173290, −13.15170552782942, −12.61038551855622, −12.29569207527150, −11.73125767301186, −11.18342622070389, −10.43076002985239, −10.09226995883288, −9.452755111193007, −9.191781950436390, −8.813335381159871, −8.140729718547871, −7.248008637405705, −7.077734856357380, −6.675765478513768, −5.879783134707049, −5.481980616987878, −4.695326936792723, −3.916064724320403, −3.654867572584342, −2.902004439039215, −2.614131410669411, −1.569727305364384, −0.9592956656380533, 0, 0.9592956656380533, 1.569727305364384, 2.614131410669411, 2.902004439039215, 3.654867572584342, 3.916064724320403, 4.695326936792723, 5.481980616987878, 5.879783134707049, 6.675765478513768, 7.077734856357380, 7.248008637405705, 8.140729718547871, 8.813335381159871, 9.191781950436390, 9.452755111193007, 10.09226995883288, 10.43076002985239, 11.18342622070389, 11.73125767301186, 12.29569207527150, 12.61038551855622, 13.15170552782942, 13.57769385173290, 14.07330232599005

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