Properties

Degree 2
Conductor $ 2^{6} \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  + 2·7-s − 11-s + 4·13-s + 6·17-s + 4·19-s − 6·23-s − 5·25-s + 6·29-s + 8·31-s + 10·37-s − 6·41-s − 8·43-s + 6·47-s − 3·49-s − 8·61-s + 4·67-s − 6·71-s + 2·73-s − 2·77-s + 14·79-s − 12·83-s + 6·89-s + 8·91-s + 14·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 0.755·7-s − 0.301·11-s + 1.10·13-s + 1.45·17-s + 0.917·19-s − 1.25·23-s − 25-s + 1.11·29-s + 1.43·31-s + 1.64·37-s − 0.937·41-s − 1.21·43-s + 0.875·47-s − 3/7·49-s − 1.02·61-s + 0.488·67-s − 0.712·71-s + 0.234·73-s − 0.227·77-s + 1.57·79-s − 1.31·83-s + 0.635·89-s + 0.838·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6336\)    =    \(2^{6} \cdot 3^{2} \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6336} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6336,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.553217758$
$L(\frac12)$  $\approx$  $2.553217758$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−17.45683041002158, −16.71242541695327, −16.08358634644137, −15.68303281495267, −14.97993722759316, −14.27711184943757, −13.73425043140939, −13.43075660294492, −12.36818960439359, −11.79402683985652, −11.52418590126793, −10.54662147099350, −10.05034700525451, −9.499557191771486, −8.433617471091069, −8.023485538763250, −7.621982552968560, −6.466985288981189, −5.932193368970334, −5.207038191223668, −4.433178300980008, −3.606174926328186, −2.833721924735947, −1.691954351535712, −0.8952075277655702, 0.8952075277655702, 1.691954351535712, 2.833721924735947, 3.606174926328186, 4.433178300980008, 5.207038191223668, 5.932193368970334, 6.466985288981189, 7.621982552968560, 8.023485538763250, 8.433617471091069, 9.499557191771486, 10.05034700525451, 10.54662147099350, 11.52418590126793, 11.79402683985652, 12.36818960439359, 13.43075660294492, 13.73425043140939, 14.27711184943757, 14.97993722759316, 15.68303281495267, 16.08358634644137, 16.71242541695327, 17.45683041002158

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