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Properties

Label 2-11-11.10-c8-0-1
Degree $2$
Conductor $11$
Sign $-0.701 + 0.712i$
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  + 31.0i·2-s − 35.5·3-s − 708.·4-s + 583.·5-s − 1.10e3i·6-s + 1.70e3i·7-s − 1.40e4i·8-s − 5.29e3·9-s + 1.81e4i·10-s + (−1.02e4 + 1.04e4i)11-s + 2.51e4·12-s + 463. i·13-s − 5.29e4·14-s − 2.07e4·15-s + 2.55e5·16-s + 4.70e4i·17-s + ⋯
L(s)  = 1  + 1.94i·2-s − 0.438·3-s − 2.76·4-s + 0.932·5-s − 0.851i·6-s + 0.710i·7-s − 3.43i·8-s − 0.807·9-s + 1.81i·10-s + (−0.701 + 0.712i)11-s + 1.21·12-s + 0.0162i·13-s − 1.37·14-s − 0.409·15-s + 3.89·16-s + 0.562i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11\)
Sign: $-0.701 + 0.712i$
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: \((2,\ 11,\ (\ :4),\ -0.701 + 0.712i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.338592 - 0.808949i\)
\(L(\frac12)\) \(\approx\) \(0.338592 - 0.808949i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (1.02e4 - 1.04e4i)T \)
good2 \( 1 - 31.0iT - 256T^{2} \)
3 \( 1 + 35.5T + 6.56e3T^{2} \)
5 \( 1 - 583.T + 3.90e5T^{2} \)
7 \( 1 - 1.70e3iT - 5.76e6T^{2} \)
13 \( 1 - 463. iT - 8.15e8T^{2} \)
17 \( 1 - 4.70e4iT - 6.97e9T^{2} \)
19 \( 1 - 1.93e5iT - 1.69e10T^{2} \)
23 \( 1 - 3.28e5T + 7.83e10T^{2} \)
29 \( 1 + 3.78e5iT - 5.00e11T^{2} \)
31 \( 1 - 5.31e5T + 8.52e11T^{2} \)
37 \( 1 - 1.49e5T + 3.51e12T^{2} \)
41 \( 1 - 3.07e6iT - 7.98e12T^{2} \)
43 \( 1 + 3.56e5iT - 1.16e13T^{2} \)
47 \( 1 + 3.12e6T + 2.38e13T^{2} \)
53 \( 1 + 5.30e6T + 6.22e13T^{2} \)
59 \( 1 - 6.23e6T + 1.46e14T^{2} \)
61 \( 1 + 2.27e7iT - 1.91e14T^{2} \)
67 \( 1 - 1.47e7T + 4.06e14T^{2} \)
71 \( 1 - 1.38e7T + 6.45e14T^{2} \)
73 \( 1 - 1.89e7iT - 8.06e14T^{2} \)
79 \( 1 - 2.11e7iT - 1.51e15T^{2} \)
83 \( 1 - 2.39e7iT - 2.25e15T^{2} \)
89 \( 1 + 6.54e7T + 3.93e15T^{2} \)
97 \( 1 - 5.14e7T + 7.83e15T^{2} \)
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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

−18.64309820398201915653042501142, −17.60468491547697858209016483427, −16.80416927803294489189252164067, −15.34681729444980199346591024664, −14.23371184435340827634643509424, −12.79335841059255648331797888300, −9.764707497223607902641901050069, −8.218550649168762837333633277563, −6.24510921386623693170860302236, −5.25917659799579852437183915228, 0.62052206104664470340057151047, 2.79562973776196031866397220009, 5.16451808118839730554324339454, 8.991405619870339410085558409419, 10.48737130850044921602211883026, 11.41298592110901873373367219038, 13.16586922153758465367499939748, 13.95805434567369653603889498057, 17.13775475499411864240348470983, 17.99214487280421453135629541406

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