Properties

Degree 2
Conductor $ 3 \cdot 11^{2} \cdot 13^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·4-s − 2·5-s + 7-s + 9-s + 2·12-s + 2·15-s + 4·16-s − 2·17-s − 4·19-s + 4·20-s − 21-s + 4·23-s − 25-s − 27-s − 2·28-s + 8·29-s + 3·31-s − 2·35-s − 2·36-s − 10·37-s − 8·41-s + 7·43-s − 2·45-s − 8·47-s − 4·48-s − 6·49-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s − 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.577·12-s + 0.516·15-s + 16-s − 0.485·17-s − 0.917·19-s + 0.894·20-s − 0.218·21-s + 0.834·23-s − 1/5·25-s − 0.192·27-s − 0.377·28-s + 1.48·29-s + 0.538·31-s − 0.338·35-s − 1/3·36-s − 1.64·37-s − 1.24·41-s + 1.06·43-s − 0.298·45-s − 1.16·47-s − 0.577·48-s − 6/7·49-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  =  \(0\)
Selberg data  =  \((2,\ 61347,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.1935792586\)
\(L(\frac12)\)  \(\approx\)  \(0.1935792586\)
\(L(\frac{3}{2})\)   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 \( 1 + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 - 7 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−14.25983021848070, −13.72008218246545, −13.33223981898584, −12.60710715500261, −12.32412851675405, −11.83573171429198, −11.24634233728024, −10.72365168489377, −10.32797933737857, −9.713596062235033, −9.033425020329383, −8.564952194487227, −8.173578301283828, −7.615153125150030, −6.967117165000267, −6.366477511852332, −5.846572834565879, −4.951487137658432, −4.601427942628429, −4.403776894738268, −3.407293849972879, −3.086193477772935, −1.853314856234728, −1.193592273015639, −0.1734224638246669, 0.1734224638246669, 1.193592273015639, 1.853314856234728, 3.086193477772935, 3.407293849972879, 4.403776894738268, 4.601427942628429, 4.951487137658432, 5.846572834565879, 6.366477511852332, 6.967117165000267, 7.615153125150030, 8.173578301283828, 8.564952194487227, 9.033425020329383, 9.713596062235033, 10.32797933737857, 10.72365168489377, 11.24634233728024, 11.83573171429198, 12.32412851675405, 12.60710715500261, 13.33223981898584, 13.72008218246545, 14.25983021848070

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