Properties

Label 2-11-1.1-c3-0-0
Degree $2$
Conductor $11$
Sign $1$
Analytic cond. $0.649021$
Root an. cond. $0.805618$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.732·2-s + 5.92·3-s − 7.46·4-s − 12.8·5-s − 4.33·6-s + 16.9·7-s + 11.3·8-s + 8.14·9-s + 9.41·10-s − 11·11-s − 44.2·12-s + 74.6·13-s − 12.3·14-s − 76.2·15-s + 51.4·16-s − 82.7·17-s − 5.96·18-s − 67.9·19-s + 95.9·20-s + 100.·21-s + 8.05·22-s + 13.3·23-s + 67.1·24-s + 40.2·25-s − 54.6·26-s − 111.·27-s − 126.·28-s + ⋯
L(s)  = 1  − 0.258·2-s + 1.14·3-s − 0.933·4-s − 1.14·5-s − 0.295·6-s + 0.914·7-s + 0.500·8-s + 0.301·9-s + 0.297·10-s − 0.301·11-s − 1.06·12-s + 1.59·13-s − 0.236·14-s − 1.31·15-s + 0.803·16-s − 1.18·17-s − 0.0780·18-s − 0.820·19-s + 1.07·20-s + 1.04·21-s + 0.0780·22-s + 0.121·23-s + 0.570·24-s + 0.322·25-s − 0.412·26-s − 0.796·27-s − 0.852·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11\)
Sign: $1$
Analytic conductor: \(0.649021\)
Root analytic conductor: \(0.805618\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{11} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 11,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8560195510\)
\(L(\frac12)\) \(\approx\) \(0.8560195510\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + 11T \)
good2 \( 1 + 0.732T + 8T^{2} \)
3 \( 1 - 5.92T + 27T^{2} \)
5 \( 1 + 12.8T + 125T^{2} \)
7 \( 1 - 16.9T + 343T^{2} \)
13 \( 1 - 74.6T + 2.19e3T^{2} \)
17 \( 1 + 82.7T + 4.91e3T^{2} \)
19 \( 1 + 67.9T + 6.85e3T^{2} \)
23 \( 1 - 13.3T + 1.21e4T^{2} \)
29 \( 1 - 168.T + 2.43e4T^{2} \)
31 \( 1 + 65.4T + 2.97e4T^{2} \)
37 \( 1 - 40.8T + 5.06e4T^{2} \)
41 \( 1 - 274.T + 6.89e4T^{2} \)
43 \( 1 + 2.28T + 7.95e4T^{2} \)
47 \( 1 - 71.8T + 1.03e5T^{2} \)
53 \( 1 + 149.T + 1.48e5T^{2} \)
59 \( 1 - 545.T + 2.05e5T^{2} \)
61 \( 1 - 101.T + 2.26e5T^{2} \)
67 \( 1 - 411.T + 3.00e5T^{2} \)
71 \( 1 + 470.T + 3.57e5T^{2} \)
73 \( 1 - 610.T + 3.89e5T^{2} \)
79 \( 1 + 978.T + 4.93e5T^{2} \)
83 \( 1 - 26.1T + 5.71e5T^{2} \)
89 \( 1 + 352.T + 7.04e5T^{2} \)
97 \( 1 - 847.T + 9.12e5T^{2} \)
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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

−19.98272888814166802692057074899, −18.95082872831269086040641176420, −17.74965087372963331749051957749, −15.70426635999610925768034563766, −14.45413212802334498598783244085, −13.23394761241722868999602498309, −11.08380472054021309562065233191, −8.738362642763152216189473172240, −8.090000841979261224322862516068, −4.10903776582739696739959352561, 4.10903776582739696739959352561, 8.090000841979261224322862516068, 8.738362642763152216189473172240, 11.08380472054021309562065233191, 13.23394761241722868999602498309, 14.45413212802334498598783244085, 15.70426635999610925768034563766, 17.74965087372963331749051957749, 18.95082872831269086040641176420, 19.98272888814166802692057074899

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