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Properties

Label 2-11-11.10-c12-0-9
Degree $2$
Conductor $11$
Sign $-0.789 + 0.613i$
Analytic cond. $10.0539$
Root an. cond. $3.17079$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 110. i·2-s + 1.25e3·3-s − 8.01e3·4-s + 1.40e4·5-s − 1.38e5i·6-s − 1.31e5i·7-s + 4.31e5i·8-s + 1.05e6·9-s − 1.54e6i·10-s + (−1.39e6 + 1.08e6i)11-s − 1.00e7·12-s − 3.22e6i·13-s − 1.44e7·14-s + 1.76e7·15-s + 1.46e7·16-s + 3.44e7i·17-s + ⋯
L(s)  = 1  − 1.71i·2-s + 1.72·3-s − 1.95·4-s + 0.896·5-s − 2.96i·6-s − 1.11i·7-s + 1.64i·8-s + 1.98·9-s − 1.54i·10-s + (−0.789 + 0.613i)11-s − 3.37·12-s − 0.667i·13-s − 1.91·14-s + 1.54·15-s + 0.875·16-s + 1.42i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.789 + 0.613i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.789 + 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11\)
Sign: $-0.789 + 0.613i$
Analytic conductor: \(10.0539\)
Root analytic conductor: \(3.17079\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{11} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 11,\ (\ :6),\ -0.789 + 0.613i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.964878 - 2.81422i\)
\(L(\frac12)\) \(\approx\) \(0.964878 - 2.81422i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (1.39e6 - 1.08e6i)T \)
good2 \( 1 + 110. iT - 4.09e3T^{2} \)
3 \( 1 - 1.25e3T + 5.31e5T^{2} \)
5 \( 1 - 1.40e4T + 2.44e8T^{2} \)
7 \( 1 + 1.31e5iT - 1.38e10T^{2} \)
13 \( 1 + 3.22e6iT - 2.32e13T^{2} \)
17 \( 1 - 3.44e7iT - 5.82e14T^{2} \)
19 \( 1 - 3.46e7iT - 2.21e15T^{2} \)
23 \( 1 - 1.65e8T + 2.19e16T^{2} \)
29 \( 1 + 9.11e8iT - 3.53e17T^{2} \)
31 \( 1 - 2.56e8T + 7.87e17T^{2} \)
37 \( 1 - 1.07e9T + 6.58e18T^{2} \)
41 \( 1 - 5.70e9iT - 2.25e19T^{2} \)
43 \( 1 - 3.44e9iT - 3.99e19T^{2} \)
47 \( 1 + 2.28e9T + 1.16e20T^{2} \)
53 \( 1 - 6.37e9T + 4.91e20T^{2} \)
59 \( 1 - 4.47e10T + 1.77e21T^{2} \)
61 \( 1 + 2.20e9iT - 2.65e21T^{2} \)
67 \( 1 + 3.46e10T + 8.18e21T^{2} \)
71 \( 1 + 1.37e11T + 1.64e22T^{2} \)
73 \( 1 - 6.76e10iT - 2.29e22T^{2} \)
79 \( 1 - 9.62e10iT - 5.90e22T^{2} \)
83 \( 1 + 3.17e11iT - 1.06e23T^{2} \)
89 \( 1 - 1.40e11T + 2.46e23T^{2} \)
97 \( 1 + 1.24e12T + 6.93e23T^{2} \)
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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

−17.57712457349627132441267789454, −14.83381223059906975971224692832, −13.45450885619099553642762729114, −13.00313800695138363655781012046, −10.39694214978782415241776048769, −9.733568124363038645427389214248, −8.050546725184329346009239156189, −4.06449061974704228475607615387, −2.67005105637449006321236845487, −1.44475219909566298410972226948, 2.61142004654666547042672433941, 5.25060057668144635018352723148, 7.12101329862969705465991007262, 8.700815823306516410150167703658, 9.297271228717262355226873169731, 13.31485858651750151166195224677, 14.07569937092332758962514612545, 15.17570663518177245441128139324, 16.14838843608979842208771014690, 18.04896210613253060792344479182

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