Properties

Degree 2
Conductor $ 3 \cdot 11^{2} \cdot 13^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3-s + 2·4-s − 4·5-s + 2·6-s + 7-s + 9-s + 8·10-s − 2·12-s − 2·14-s + 4·15-s − 4·16-s − 4·17-s − 2·18-s − 3·19-s − 8·20-s − 21-s + 2·23-s + 11·25-s − 27-s + 2·28-s − 6·29-s − 8·30-s + 5·31-s + 8·32-s + 8·34-s − 4·35-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 4-s − 1.78·5-s + 0.816·6-s + 0.377·7-s + 1/3·9-s + 2.52·10-s − 0.577·12-s − 0.534·14-s + 1.03·15-s − 16-s − 0.970·17-s − 0.471·18-s − 0.688·19-s − 1.78·20-s − 0.218·21-s + 0.417·23-s + 11/5·25-s − 0.192·27-s + 0.377·28-s − 1.11·29-s − 1.46·30-s + 0.898·31-s + 1.41·32-s + 1.37·34-s − 0.676·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61347 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61347 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(61347\)    =    \(3 \cdot 11^{2} \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{61347} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(2\)
Selberg data  =  \((2,\ 61347,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;11,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;11,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + p T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−15.04688067512794, −14.62769076761707, −13.64740303276649, −13.17052504705317, −12.61980417905522, −11.96294900305826, −11.41910570697875, −11.30289728745219, −10.84269175602116, −10.14382234172200, −9.832818210881621, −8.821049253127353, −8.593446917283691, −8.276956624108366, −7.518152042162350, −7.203838870766306, −6.740008047067660, −6.107600204653647, −4.981716473821834, −4.741552463920491, −4.061142771192423, −3.515162729081230, −2.579031215256743, −1.743729842763260, −1.037541007206339, 0, 0, 1.037541007206339, 1.743729842763260, 2.579031215256743, 3.515162729081230, 4.061142771192423, 4.741552463920491, 4.981716473821834, 6.107600204653647, 6.740008047067660, 7.203838870766306, 7.518152042162350, 8.276956624108366, 8.593446917283691, 8.821049253127353, 9.832818210881621, 10.14382234172200, 10.84269175602116, 11.30289728745219, 11.41910570697875, 11.96294900305826, 12.61980417905522, 13.17052504705317, 13.64740303276649, 14.62769076761707, 15.04688067512794

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