Properties

Degree 2
Conductor 11
Sign $1$
Self-dual yes
Motivic weight 1

Origins

Related objects

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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  − 1.414·2-s − 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s − 0.755·7-s − 0.666·9-s − 0.632·10-s + 0.301·11-s − 0.577·12-s + 1.109·13-s + 1.069·14-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 − 0.8·25-s − 1.568·26-s + 0.962·27-s − 0.755·28-s + 0.365·30-s + 1.257·31-s + 1.414·32-s + ⋯

Functional equation

\[\begin{align} \Lambda(s)=\mathstrut & 11 ^{s/2} \Gamma_{\C}(s) \cdot L(s)\cr =\mathstrut & \Lambda(2-s) \end{align} \]
\[\begin{align} \Lambda(s)=\mathstrut & 11 ^{s/2} \Gamma_{\C}(s+0.5) \cdot L(s)\cr =\mathstrut & \Lambda(1-s) \end{align} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(11\)
\( \varepsilon \)  =  $1$
weight  =  1
character  :  $\chi_{11} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 11,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.2538418609$
$L(\frac12)$  $\approx$  $0.2538418609$
$L(\frac{3}{2})$   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$1+2T+2T^{2}$
3$1+T+3T^{2}$
5$1-T+5T^{2}$
7$1+2T+7T^{2}$
13$1-4T+13T^{2}$
17$1+2T+17T^{2}$
19$1+19T^{2}$
23$1+T+23T^{2}$
29$1+29T^{2}$
31$1-7T+31T^{2}$
37$1-3T+37T^{2}$
41$1+8T+41T^{2}$
43$1+6T+43T^{2}$
47$1-8T+47T^{2}$
53$1+6T+53T^{2}$
59$1-5T+59T^{2}$
61$1-12T+61T^{2}$
67$1+7T+67T^{2}$
71$1+3T+71T^{2}$
73$1-4T+73T^{2}$
79$1+10T+79T^{2}$
83$1+6T+83T^{2}$
89$1-15T+89T^{2}$
97$1+7T+97T^{2}$
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\[\begin{equation} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{equation}\]

Imaginary part of the first few zeros on the critical line

−19.18572497185224, −17.94143357345934, −17.03361032038062, −15.91407260330038, −13.56863905712999, −11.45125861034521, −10.03550909718108, −8.603539619290756, −6.362613894713089, 6.362613894713089, 8.603539619290756, 10.03550909718108, 11.45125861034521, 13.56863905712999, 15.91407260330038, 17.03361032038062, 17.94143357345934, 19.18572497185224

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