Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5^{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  − 3-s + 4·7-s + 9-s − 11-s + 2·13-s + 2·17-s − 4·21-s + 8·23-s − 27-s − 6·29-s + 8·31-s + 33-s − 6·37-s − 2·39-s − 2·41-s + 8·47-s + 9·49-s − 2·51-s − 6·53-s + 4·59-s + 6·61-s + 4·63-s − 4·67-s − 8·69-s + 14·73-s − 4·77-s + 4·79-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.51·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.485·17-s − 0.872·21-s + 1.66·23-s − 0.192·27-s − 1.11·29-s + 1.43·31-s + 0.174·33-s − 0.986·37-s − 0.320·39-s − 0.312·41-s + 1.16·47-s + 9/7·49-s − 0.280·51-s − 0.824·53-s + 0.520·59-s + 0.768·61-s + 0.503·63-s − 0.488·67-s − 0.963·69-s + 1.63·73-s − 0.455·77-s + 0.450·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(13200\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{13200} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 13200,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.454874704$
$L(\frac12)$  $\approx$  $2.454874704$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
11 \( 1 + T \)
good7 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−16.27256062614275, −15.62457197946875, −15.06436324433283, −14.73119084372890, −13.83215295509025, −13.60110345583150, −12.72788500880245, −12.23509685811734, −11.52908329007567, −11.13995369938861, −10.69809771120846, −10.07326435159755, −9.232479056657205, −8.617395750063261, −8.035493206994264, −7.454602592603183, −6.814179386520870, −6.041608919261256, −5.138214846379046, −5.095965669978320, −4.169487281660123, −3.380527732613603, −2.366949521923393, −1.483468539209140, −0.7946702187462776, 0.7946702187462776, 1.483468539209140, 2.366949521923393, 3.380527732613603, 4.169487281660123, 5.095965669978320, 5.138214846379046, 6.041608919261256, 6.814179386520870, 7.454602592603183, 8.035493206994264, 8.617395750063261, 9.232479056657205, 10.07326435159755, 10.69809771120846, 11.13995369938861, 11.52908329007567, 12.23509685811734, 12.72788500880245, 13.60110345583150, 13.83215295509025, 14.73119084372890, 15.06436324433283, 15.62457197946875, 16.27256062614275

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