Properties

Degree $2$
Conductor $2112$
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 + 6·13-s + 2·15-s + 2·17-s + 4·19-s − 4·21-s − 4·23-s − 25-s − 27-s − 6·29-s + 33-s − 8·35-s − 6·37-s − 6·39-s − 6·41-s + 4·43-s − 2·45-s + 12·47-s + 9·49-s − 2·51-s − 2·53-s + 2·55-s − 4·57-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 1.51·7-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 − 0.872·21-s − 0.834·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.174·33-s − 1.35·35-s − 0.986·37-s − 0.960·39-s − 0.937·41-s + 0.609·43-s − 0.298·45-s + 1.75·47-s + 9/7·49-s − 0.280·51-s − 0.274·53-s + 0.269·55-s − 0.529·57-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

Degree: \(2\)
Conductor: \(2112\)    =    \(2^{6} \cdot 3 \cdot 11\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{2112} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2112,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.551434499\)
\(L(\frac12)\) \(\approx\) \(1.551434499\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
11 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
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 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.881392037510841834891251946908, −8.149492705426918418612295849228, −7.72333700605509306004633046648, −6.86624573821975866794841152731, −5.64980027395210574108319205224, −5.28231085552121772501988818495, −4.09502695849124324670131106925, −3.63086619141261654419365571100, −1.94521935316309572966615388289, −0.894868895370507138388762935569, 0.894868895370507138388762935569, 1.94521935316309572966615388289, 3.63086619141261654419365571100, 4.09502695849124324670131106925, 5.28231085552121772501988818495, 5.64980027395210574108319205224, 6.86624573821975866794841152731, 7.72333700605509306004633046648, 8.149492705426918418612295849228, 8.881392037510841834891251946908

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