Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \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 + 2·5-s + 9-s + 11-s + 6·13-s − 2·15-s + 2·17-s + 4·19-s − 4·23-s − 25-s − 27-s − 2·29-s − 4·31-s − 33-s + 2·37-s − 6·39-s + 10·41-s + 4·43-s + 2·45-s − 4·47-s − 7·49-s − 2·51-s − 6·53-s + 2·55-s − 4·57-s + 12·59-s + 6·61-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.301·11-s + 1.66·13-s − 0.516·15-s + 0.485·17-s + 0.917·19-s − 0.834·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.174·33-s + 0.328·37-s − 0.960·39-s + 1.56·41-s + 0.609·43-s + 0.298·45-s − 0.583·47-s − 49-s − 0.280·51-s − 0.824·53-s + 0.269·55-s − 0.529·57-s + 1.56·59-s + 0.768·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2112\)    =    \(2^{6} \cdot 3 \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{2112} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 2112,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.961990463$
$L(\frac12)$  $\approx$  $1.961990463$
$L(\frac{3}{2})$   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 \( 1 + T \)
11 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−19.15520691303461, −18.43769290665737, −17.93878233335363, −17.53790067534449, −16.68400884258914, −16.01574911707526, −15.74678925698324, −14.35965908895079, −14.23401941136567, −13.15232279774212, −12.94044448553744, −11.80515204752485, −11.31065754799906, −10.60224793656040, −9.765447944134856, −9.314470900621634, −8.329522851135257, −7.522149262537255, −6.501161925947473, −5.885964881985012, −5.420296525627633, −4.183915068278919, −3.370237254317549, −1.971593257713592, −1.025025820815528, 1.025025820815528, 1.971593257713592, 3.370237254317549, 4.183915068278919, 5.420296525627633, 5.885964881985012, 6.501161925947473, 7.522149262537255, 8.329522851135257, 9.314470900621634, 9.765447944134856, 10.60224793656040, 11.31065754799906, 11.80515204752485, 12.94044448553744, 13.15232279774212, 14.23401941136567, 14.35965908895079, 15.74678925698324, 16.01574911707526, 16.68400884258914, 17.53790067534449, 17.93878233335363, 18.43769290665737, 19.15520691303461

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