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 + 4·7-s + 9-s − 11-s + 2·13-s + 2·15-s − 2·17-s + 4·21-s + 8·23-s − 25-s + 27-s + 6·29-s − 8·31-s − 33-s + 8·35-s − 6·37-s + 2·39-s − 2·41-s + 2·45-s + 8·47-s + 9·49-s − 2·51-s − 6·53-s − 2·55-s + 4·59-s − 6·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 1.51·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.516·15-s − 0.485·17-s + 0.872·21-s + 1.66·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.174·33-s + 1.35·35-s − 0.986·37-s + 0.320·39-s − 0.312·41-s + 0.298·45-s + 1.16·47-s + 9/7·49-s − 0.280·51-s − 0.824·53-s − 0.269·55-s + 0.520·59-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  =  $(2,\ 2112,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.170691475$
$L(\frac12)$  $\approx$  $3.170691475$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 - T \)
11 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
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

−19.25652277678583, −18.49274219178283, −18.05328875766157, −17.37273024751971, −16.99286123222253, −15.85282406616695, −15.34586041848402, −14.51693292301889, −14.15865683457026, −13.44100353735610, −12.92009656555026, −11.95833266553108, −11.05629761230732, −10.68692529121914, −9.785147651821030, −8.815632295433107, −8.598193922290111, −7.577669022135304, −6.913425435185022, −5.795082551062696, −5.089119557656205, −4.325268817556292, −3.108633847078939, −2.075520524057637, −1.331481515753509, 1.331481515753509, 2.075520524057637, 3.108633847078939, 4.325268817556292, 5.089119557656205, 5.795082551062696, 6.913425435185022, 7.577669022135304, 8.598193922290111, 8.815632295433107, 9.785147651821030, 10.68692529121914, 11.05629761230732, 11.95833266553108, 12.92009656555026, 13.44100353735610, 14.15865683457026, 14.51693292301889, 15.34586041848402, 15.85282406616695, 16.99286123222253, 17.37273024751971, 18.05328875766157, 18.49274219178283, 19.25652277678583

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