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  =  $(2,\ 6336,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.952467348$
$L(\frac12)$  $\approx$  $1.952467348$
$L(\frac{3}{2})$   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

−17.17066506217223, −16.77447793132669, −16.14982815072257, −15.71597816552698, −14.87054602513677, −14.48930138839505, −13.70373776821290, −13.05475212235579, −12.70347683853923, −11.92965200142483, −11.26567896936209, −10.65404352967375, −9.995545339477385, −9.337741184325048, −8.788523562921076, −7.990007746064461, −7.383771746585810, −6.408644341015616, −6.114851730722257, −5.262854706990188, −4.306033077454239, −3.523944361257087, −2.973667419627950, −1.742896288067604, −0.7484165518892625, 0.7484165518892625, 1.742896288067604, 2.973667419627950, 3.523944361257087, 4.306033077454239, 5.262854706990188, 6.114851730722257, 6.408644341015616, 7.383771746585810, 7.990007746064461, 8.788523562921076, 9.337741184325048, 9.995545339477385, 10.65404352967375, 11.26567896936209, 11.92965200142483, 12.70347683853923, 13.05475212235579, 13.70373776821290, 14.48930138839505, 14.87054602513677, 15.71597816552698, 16.14982815072257, 16.77447793132669, 17.17066506217223

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