Properties

Label 2-11-11.10-c12-0-6
Degree $2$
Conductor $11$
Sign $0.970 + 0.239i$
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  − 28.3i·2-s + 796.·3-s + 3.29e3·4-s + 5.72e3·5-s − 2.26e4i·6-s + 2.02e5i·7-s − 2.09e5i·8-s + 1.03e5·9-s − 1.62e5i·10-s + (1.72e6 + 4.24e5i)11-s + 2.62e6·12-s − 6.56e6i·13-s + 5.74e6·14-s + 4.55e6·15-s + 7.53e6·16-s + 2.59e7i·17-s + ⋯
L(s)  = 1  − 0.443i·2-s + 1.09·3-s + 0.803·4-s + 0.366·5-s − 0.484i·6-s + 1.72i·7-s − 0.799i·8-s + 0.195·9-s − 0.162i·10-s + (0.970 + 0.239i)11-s + 0.878·12-s − 1.36i·13-s + 0.763·14-s + 0.400·15-s + 0.448·16-s + 1.07i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11\)
Sign: $0.970 + 0.239i$
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: \((2,\ 11,\ (\ :6),\ 0.970 + 0.239i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(3.02093 - 0.367053i\)
\(L(\frac12)\) \(\approx\) \(3.02093 - 0.367053i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-1.72e6 - 4.24e5i)T \)
good2 \( 1 + 28.3iT - 4.09e3T^{2} \)
3 \( 1 - 796.T + 5.31e5T^{2} \)
5 \( 1 - 5.72e3T + 2.44e8T^{2} \)
7 \( 1 - 2.02e5iT - 1.38e10T^{2} \)
13 \( 1 + 6.56e6iT - 2.32e13T^{2} \)
17 \( 1 - 2.59e7iT - 5.82e14T^{2} \)
19 \( 1 + 7.15e7iT - 2.21e15T^{2} \)
23 \( 1 + 9.33e7T + 2.19e16T^{2} \)
29 \( 1 + 6.46e7iT - 3.53e17T^{2} \)
31 \( 1 - 5.07e8T + 7.87e17T^{2} \)
37 \( 1 - 5.29e8T + 6.58e18T^{2} \)
41 \( 1 - 4.97e9iT - 2.25e19T^{2} \)
43 \( 1 + 1.97e9iT - 3.99e19T^{2} \)
47 \( 1 + 1.74e10T + 1.16e20T^{2} \)
53 \( 1 + 1.99e10T + 4.91e20T^{2} \)
59 \( 1 - 2.01e10T + 1.77e21T^{2} \)
61 \( 1 - 5.39e10iT - 2.65e21T^{2} \)
67 \( 1 - 7.88e10T + 8.18e21T^{2} \)
71 \( 1 - 4.09e10T + 1.64e22T^{2} \)
73 \( 1 + 2.05e10iT - 2.29e22T^{2} \)
79 \( 1 + 9.65e10iT - 5.90e22T^{2} \)
83 \( 1 + 2.46e11iT - 1.06e23T^{2} \)
89 \( 1 + 1.49e11T + 2.46e23T^{2} \)
97 \( 1 - 5.12e11T + 6.93e23T^{2} \)
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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

−17.66762879692539700004732699198, −15.52177107227388784100130885295, −14.85571342406667155643891900682, −12.93473329103268348797840565814, −11.60739584068539514869174042177, −9.648463989246751415028494179167, −8.292872737519272689304021340976, −6.06503730892599228626762514641, −3.05489240101678304621905459590, −1.98853782769296521458150469022, 1.73080062762996615624344506108, 3.75394629712621173613568732013, 6.60415488300852548599730045570, 7.896147381218841504840845916474, 9.735823892903834970446721576369, 11.51865714939836694291215432498, 14.05729196748031930650246394085, 14.20198259297611738686251916246, 16.26032134773203303511638247806, 17.12832332032380435712712624494

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