Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 4·7-s − 4·11-s − 2·13-s + 6·17-s − 4·19-s − 25-s − 2·29-s + 4·31-s + 8·35-s − 2·37-s − 2·41-s + 4·43-s − 8·47-s + 9·49-s − 10·53-s + 8·55-s + 4·59-s + 6·61-s + 4·65-s + 4·67-s + 16·71-s − 6·73-s + 16·77-s + 4·79-s − 12·83-s − 12·85-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.51·7-s − 1.20·11-s − 0.554·13-s + 1.45·17-s − 0.917·19-s − 1/5·25-s − 0.371·29-s + 0.718·31-s + 1.35·35-s − 0.328·37-s − 0.312·41-s + 0.609·43-s − 1.16·47-s + 9/7·49-s − 1.37·53-s + 1.07·55-s + 0.520·59-s + 0.768·61-s + 0.496·65-s + 0.488·67-s + 1.89·71-s − 0.702·73-s + 1.82·77-s + 0.450·79-s − 1.31·83-s − 1.30·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(288\)    =    \(2^{5} \cdot 3^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{288} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 288,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

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

−11.39516795962572483098752274149, −10.23151567133484212790849208076, −9.689914794558097820657325082483, −8.291996087541312922253124488251, −7.52219997040636286950635023716, −6.44434197467298206476021389307, −5.25332741283121206700463308290, −3.80880328694342425730935973216, −2.80335944085781759882923856447, 0, 2.80335944085781759882923856447, 3.80880328694342425730935973216, 5.25332741283121206700463308290, 6.44434197467298206476021389307, 7.52219997040636286950635023716, 8.291996087541312922253124488251, 9.689914794558097820657325082483, 10.23151567133484212790849208076, 11.39516795962572483098752274149

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