Properties

Label 2-11-11.6-c12-0-8
Degree $2$
Conductor $11$
Sign $-0.687 - 0.726i$
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  + (−79.4 + 25.8i)2-s + (−641. − 466. i)3-s + (2.33e3 − 1.69e3i)4-s + (6.73e3 − 2.07e4i)5-s + (6.30e4 + 2.04e4i)6-s + (−2.48e4 − 3.41e4i)7-s + (5.96e4 − 8.21e4i)8-s + (3.01e4 + 9.27e4i)9-s + 1.82e6i·10-s + (−1.53e6 − 8.81e5i)11-s − 2.28e6·12-s + (−4.87e6 + 1.58e6i)13-s + (2.85e6 + 2.07e6i)14-s + (−1.39e7 + 1.01e7i)15-s + (−6.26e6 + 1.92e7i)16-s + (1.85e7 + 6.03e6i)17-s + ⋯
L(s)  = 1  + (−1.24 + 0.403i)2-s + (−0.880 − 0.639i)3-s + (0.568 − 0.413i)4-s + (0.431 − 1.32i)5-s + (1.35 + 0.438i)6-s + (−0.211 − 0.290i)7-s + (0.227 − 0.313i)8-s + (0.0567 + 0.174i)9-s + 1.82i·10-s + (−0.867 − 0.497i)11-s − 0.765·12-s + (−1.01 + 0.328i)13-s + (0.379 + 0.275i)14-s + (−1.22 + 0.891i)15-s + (−0.373 + 1.14i)16-s + (0.770 + 0.250i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.687 - 0.726i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11\)
Sign: $-0.687 - 0.726i$
Analytic conductor: \(10.0539\)
Root analytic conductor: \(3.17079\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{11} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 11,\ (\ :6),\ -0.687 - 0.726i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.0481809 + 0.111948i\)
\(L(\frac12)\) \(\approx\) \(0.0481809 + 0.111948i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (1.53e6 + 8.81e5i)T \)
good2 \( 1 + (79.4 - 25.8i)T + (3.31e3 - 2.40e3i)T^{2} \)
3 \( 1 + (641. + 466. i)T + (1.64e5 + 5.05e5i)T^{2} \)
5 \( 1 + (-6.73e3 + 2.07e4i)T + (-1.97e8 - 1.43e8i)T^{2} \)
7 \( 1 + (2.48e4 + 3.41e4i)T + (-4.27e9 + 1.31e10i)T^{2} \)
13 \( 1 + (4.87e6 - 1.58e6i)T + (1.88e13 - 1.36e13i)T^{2} \)
17 \( 1 + (-1.85e7 - 6.03e6i)T + (4.71e14 + 3.42e14i)T^{2} \)
19 \( 1 + (-3.34e7 + 4.59e7i)T + (-6.83e14 - 2.10e15i)T^{2} \)
23 \( 1 + 9.37e7T + 2.19e16T^{2} \)
29 \( 1 + (-3.56e8 - 4.90e8i)T + (-1.09e17 + 3.36e17i)T^{2} \)
31 \( 1 + (-1.38e7 - 4.27e7i)T + (-6.37e17 + 4.62e17i)T^{2} \)
37 \( 1 + (3.47e9 - 2.52e9i)T + (2.03e18 - 6.26e18i)T^{2} \)
41 \( 1 + (1.73e9 - 2.38e9i)T + (-6.97e18 - 2.14e19i)T^{2} \)
43 \( 1 + 9.17e9iT - 3.99e19T^{2} \)
47 \( 1 + (-6.98e9 - 5.07e9i)T + (3.59e19 + 1.10e20i)T^{2} \)
53 \( 1 + (-6.37e9 - 1.96e10i)T + (-3.97e20 + 2.88e20i)T^{2} \)
59 \( 1 + (-4.52e10 + 3.28e10i)T + (5.49e20 - 1.69e21i)T^{2} \)
61 \( 1 + (-7.68e10 - 2.49e10i)T + (2.14e21 + 1.56e21i)T^{2} \)
67 \( 1 + 1.16e11T + 8.18e21T^{2} \)
71 \( 1 + (3.02e10 - 9.30e10i)T + (-1.32e22 - 9.64e21i)T^{2} \)
73 \( 1 + (7.51e10 + 1.03e11i)T + (-7.07e21 + 2.17e22i)T^{2} \)
79 \( 1 + (3.24e11 - 1.05e11i)T + (4.78e22 - 3.47e22i)T^{2} \)
83 \( 1 + (2.19e11 + 7.13e10i)T + (8.64e22 + 6.28e22i)T^{2} \)
89 \( 1 - 3.86e11T + 2.46e23T^{2} \)
97 \( 1 + (-2.61e11 - 8.05e11i)T + (-5.61e23 + 4.07e23i)T^{2} \)
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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

−16.97469097631587141227832998373, −16.06614144145178073732563712676, −13.34836172741979901035896580499, −12.13561092993102682236489436648, −10.11305222685862347392606413433, −8.699692165208071589006453153217, −7.12522250952277432555877187814, −5.30974530025670788611332362680, −1.14916757413986422679127707370, −0.11112414909293703682553498396, 2.46577636794213842268361139489, 5.46964794153647449009986102021, 7.58528111395306951355095052391, 10.05219706900375882635128057682, 10.27263980272925151422925551782, 11.80625565020351509878551936679, 14.32897209994423492342562715780, 16.00269044121587343118001819540, 17.39826781193219993928674765343, 18.21726888213166391111595483553

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