Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 11 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 2·7-s + 11-s − 2·13-s − 4·17-s + 6·19-s − 25-s + 8·29-s + 8·31-s − 4·35-s + 10·37-s − 8·41-s + 2·43-s − 8·47-s − 3·49-s + 2·53-s − 2·55-s + 12·59-s + 10·61-s + 4·65-s − 12·67-s + 8·71-s + 6·73-s + 2·77-s + 2·79-s + 16·83-s + 8·85-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.755·7-s + 0.301·11-s − 0.554·13-s − 0.970·17-s + 1.37·19-s − 1/5·25-s + 1.48·29-s + 1.43·31-s − 0.676·35-s + 1.64·37-s − 1.24·41-s + 0.304·43-s − 1.16·47-s − 3/7·49-s + 0.274·53-s − 0.269·55-s + 1.56·59-s + 1.28·61-s + 0.496·65-s − 1.46·67-s + 0.949·71-s + 0.702·73-s + 0.227·77-s + 0.225·79-s + 1.75·83-s + 0.867·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1584\)    =    \(2^{4} \cdot 3^{2} \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1584} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 1584,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.533632354$
$L(\frac12)$  $\approx$  $1.533632354$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.72832385737805, −19.46521067893648, −18.52659913084181, −17.71780645494079, −17.50175075404564, −16.34604970062099, −15.92588924508106, −15.14698274546517, −14.62469127734738, −13.80530600685942, −13.18982590149372, −11.98066779611150, −11.79700275175494, −11.13819884397501, −10.13695658731199, −9.451440439716278, −8.352307473563579, −7.995283121669507, −7.089198287686188, −6.311163456329179, −5.010520200201469, −4.526353680737417, −3.473754163335328, −2.359919060310016, −0.8903132938861897, 0.8903132938861897, 2.359919060310016, 3.473754163335328, 4.526353680737417, 5.010520200201469, 6.311163456329179, 7.089198287686188, 7.995283121669507, 8.352307473563579, 9.451440439716278, 10.13695658731199, 11.13819884397501, 11.79700275175494, 11.98066779611150, 13.18982590149372, 13.80530600685942, 14.62469127734738, 15.14698274546517, 15.92588924508106, 16.34604970062099, 17.50175075404564, 17.71780645494079, 18.52659913084181, 19.46521067893648, 19.72832385737805

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