Properties

Degree 2
Conductor 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  − 2·2-s − 3-s + 2·4-s + 5-s + 2·6-s − 2·7-s − 2·9-s − 2·10-s + 11-s − 2·12-s + 4·13-s + 4·14-s − 15-s − 4·16-s − 2·17-s + 4·18-s + 2·20-s + 2·21-s − 2·22-s − 23-s − 4·25-s − 8·26-s + 5·27-s − 4·28-s + 2·30-s + 7·31-s + 8·32-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s − 0.755·7-s − 2/3·9-s − 0.632·10-s + 0.301·11-s − 0.577·12-s + 1.10·13-s + 1.06·14-s − 0.258·15-s − 16-s − 0.485·17-s + 0.942·18-s + 0.447·20-s + 0.436·21-s − 0.426·22-s − 0.208·23-s − 4/5·25-s − 1.56·26-s + 0.962·27-s − 0.755·28-s + 0.365·30-s + 1.25·31-s + 1.41·32-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{11} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 11,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.253841$
$L(\frac12)$  $\approx$  $0.253841$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 11$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 11$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad11 \( 1 - T \)
good2 \( 1 + p T + p T^{2} \)
3 \( 1 + T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−20.37926046435010943649954977728, −19.18572497185224141236190272085, −17.94143357345934104539279635252, −17.03361032038062445390026939405, −15.91407260330038416331639924088, −13.56863905712999451342489435843, −11.45125861034521058353374083886, −10.03550909718107888433464868208, −8.603539619290756001226038948684, −6.36261389471308870138602900888, 6.36261389471308870138602900888, 8.603539619290756001226038948684, 10.03550909718107888433464868208, 11.45125861034521058353374083886, 13.56863905712999451342489435843, 15.91407260330038416331639924088, 17.03361032038062445390026939405, 17.94143357345934104539279635252, 19.18572497185224141236190272085, 20.37926046435010943649954977728

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