Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 11 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·7-s + 9-s − 11-s + 4·13-s − 6·17-s − 4·19-s − 2·21-s − 6·23-s − 5·25-s + 27-s − 6·29-s − 8·31-s − 33-s + 10·37-s + 4·39-s + 6·41-s + 8·43-s + 6·47-s − 3·49-s − 6·51-s − 4·57-s − 8·61-s − 2·63-s − 4·67-s − 6·69-s − 6·71-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s − 1.45·17-s − 0.917·19-s − 0.436·21-s − 1.25·23-s − 25-s + 0.192·27-s − 1.11·29-s − 1.43·31-s − 0.174·33-s + 1.64·37-s + 0.640·39-s + 0.937·41-s + 1.21·43-s + 0.875·47-s − 3/7·49-s − 0.840·51-s − 0.529·57-s − 1.02·61-s − 0.251·63-s − 0.488·67-s − 0.722·69-s − 0.712·71-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  =  1
Selberg data  =  $(2,\ 2112,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
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

−19.72208514084913, −19.03432202916742, −18.28819342629912, −17.91860742221965, −16.96771231736738, −16.16507208841359, −15.76409532534157, −15.15468684854275, −14.33788297487323, −13.63771175337898, −12.94394612782749, −12.74840205746634, −11.44902945290174, −10.95079690688667, −10.14772121765499, −9.258091802717213, −8.887891532747829, −7.935843343791212, −7.285273421696434, −6.171745302903457, −5.870201485683172, −4.236904006160929, −3.914253079405575, −2.685720296533940, −1.828431706423908, 0, 1.828431706423908, 2.685720296533940, 3.914253079405575, 4.236904006160929, 5.870201485683172, 6.171745302903457, 7.285273421696434, 7.935843343791212, 8.887891532747829, 9.258091802717213, 10.14772121765499, 10.95079690688667, 11.44902945290174, 12.74840205746634, 12.94394612782749, 13.63771175337898, 14.33788297487323, 15.15468684854275, 15.76409532534157, 16.16507208841359, 16.96771231736738, 17.91860742221965, 18.28819342629912, 19.03432202916742, 19.72208514084913

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