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 − 6·13-s − 2·15-s + 6·17-s + 8·19-s − 4·21-s − 25-s − 27-s + 6·29-s − 33-s + 8·35-s − 6·37-s + 6·39-s − 10·41-s + 8·43-s + 2·45-s + 9·49-s − 6·51-s − 6·53-s + 2·55-s − 8·57-s − 4·59-s + 2·61-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 1.51·7-s + 1/3·9-s + 0.301·11-s − 1.66·13-s − 0.516·15-s + 1.45·17-s + 1.83·19-s − 0.872·21-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 − 1.56·41-s + 1.21·43-s + 0.298·45-s + 9/7·49-s − 0.840·51-s − 0.824·53-s + 0.269·55-s − 1.05·57-s − 0.520·59-s + 0.256·61-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  :  $\chi_{2112} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2112,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.175303541\)
\(L(\frac12)\)  \(\approx\)  \(2.175303541\)
\(L(\frac{3}{2})\)   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 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
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

−9.339141136925654439207262533263, −8.136326692066684045105792921301, −7.56089293819005546804727949953, −6.82430943807879406400182869542, −5.56993690843411272942663805412, −5.27293036458461851477097267556, −4.57486609696928672939637526621, −3.16578625772066727003923663964, −1.96689570329397719651168220011, −1.10197409729490824142182868653, 1.10197409729490824142182868653, 1.96689570329397719651168220011, 3.16578625772066727003923663964, 4.57486609696928672939637526621, 5.27293036458461851477097267556, 5.56993690843411272942663805412, 6.82430943807879406400182869542, 7.56089293819005546804727949953, 8.136326692066684045105792921301, 9.339141136925654439207262533263

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