Properties

Label 2-11-11.10-c8-0-5
Degree $2$
Conductor $11$
Sign $-0.734 - 0.679i$
Analytic cond. $4.48116$
Root an. cond. $2.11687$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 14.5i·2-s − 66.9·3-s + 43.4·4-s − 870.·5-s + 976. i·6-s + 4.07e3i·7-s − 4.36e3i·8-s − 2.07e3·9-s + 1.26e4i·10-s + (−1.07e4 − 9.94e3i)11-s − 2.91e3·12-s − 3.71e4i·13-s + 5.94e4·14-s + 5.82e4·15-s − 5.25e4·16-s − 2.00e3i·17-s + ⋯
L(s)  = 1  − 0.911i·2-s − 0.827·3-s + 0.169·4-s − 1.39·5-s + 0.753i·6-s + 1.69i·7-s − 1.06i·8-s − 0.315·9-s + 1.26i·10-s + (−0.734 − 0.679i)11-s − 0.140·12-s − 1.29i·13-s + 1.54·14-s + 1.15·15-s − 0.801·16-s − 0.0239i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.734 - 0.679i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.734 - 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11\)
Sign: $-0.734 - 0.679i$
Analytic conductor: \(4.48116\)
Root analytic conductor: \(2.11687\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{11} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 11,\ (\ :4),\ -0.734 - 0.679i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.0387760 + 0.0990321i\)
\(L(\frac12)\) \(\approx\) \(0.0387760 + 0.0990321i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (1.07e4 + 9.94e3i)T \)
good2 \( 1 + 14.5iT - 256T^{2} \)
3 \( 1 + 66.9T + 6.56e3T^{2} \)
5 \( 1 + 870.T + 3.90e5T^{2} \)
7 \( 1 - 4.07e3iT - 5.76e6T^{2} \)
13 \( 1 + 3.71e4iT - 8.15e8T^{2} \)
17 \( 1 + 2.00e3iT - 6.97e9T^{2} \)
19 \( 1 - 8.95e4iT - 1.69e10T^{2} \)
23 \( 1 + 2.77e5T + 7.83e10T^{2} \)
29 \( 1 - 3.25e5iT - 5.00e11T^{2} \)
31 \( 1 - 4.44e5T + 8.52e11T^{2} \)
37 \( 1 + 1.48e6T + 3.51e12T^{2} \)
41 \( 1 - 1.84e6iT - 7.98e12T^{2} \)
43 \( 1 + 3.67e6iT - 1.16e13T^{2} \)
47 \( 1 - 3.89e6T + 2.38e13T^{2} \)
53 \( 1 + 6.99e6T + 6.22e13T^{2} \)
59 \( 1 + 4.71e6T + 1.46e14T^{2} \)
61 \( 1 - 7.61e6iT - 1.91e14T^{2} \)
67 \( 1 + 1.80e7T + 4.06e14T^{2} \)
71 \( 1 - 4.47e6T + 6.45e14T^{2} \)
73 \( 1 + 1.06e7iT - 8.06e14T^{2} \)
79 \( 1 - 4.97e7iT - 1.51e15T^{2} \)
83 \( 1 + 6.38e7iT - 2.25e15T^{2} \)
89 \( 1 + 1.01e7T + 3.93e15T^{2} \)
97 \( 1 + 1.28e8T + 7.83e15T^{2} \)
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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

−18.23299238857303584888674060071, −16.07884733010001424020066163746, −15.36213277962131614797666133382, −12.40646716794620736051183507494, −11.83031250380549992946213567459, −10.68172286464354837595652189654, −8.233155449778884049974857026577, −5.71929853712523623839858189954, −3.02096181626994876041925002622, −0.07219999791047698356237564162, 4.49043660283278153192048998246, 6.79246458963393192223536680385, 7.82506140638232465592995246767, 10.81778528999316642037089821459, 11.83555732644907705959175148161, 14.08457782539895700814437892284, 15.67821881314275487116669255784, 16.56626244504167105256625448072, 17.52350933248890116121992664240, 19.59062115631709699410581356251

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