Properties

Label 2-11-11.3-c11-0-5
Degree $2$
Conductor $11$
Sign $0.886 - 0.462i$
Analytic cond. $8.45177$
Root an. cond. $2.90719$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (44.3 − 32.2i)2-s + (122. + 377. i)3-s + (294. − 906. i)4-s + (3.99e3 + 2.90e3i)5-s + (1.75e4 + 1.27e4i)6-s + (−1.25e4 + 3.85e4i)7-s + (1.85e4 + 5.70e4i)8-s + (1.59e4 − 1.15e4i)9-s + 2.70e5·10-s + (2.26e5 − 4.83e5i)11-s + 3.78e5·12-s + (2.52e5 − 1.83e5i)13-s + (6.85e5 + 2.10e6i)14-s + (−6.05e5 + 1.86e6i)15-s + (4.23e6 + 3.07e6i)16-s + (−1.46e6 − 1.06e6i)17-s + ⋯
L(s)  = 1  + (0.979 − 0.711i)2-s + (0.291 + 0.896i)3-s + (0.143 − 0.442i)4-s + (0.571 + 0.415i)5-s + (0.923 + 0.670i)6-s + (−0.281 + 0.866i)7-s + (0.199 + 0.615i)8-s + (0.0900 − 0.0654i)9-s + 0.855·10-s + (0.423 − 0.905i)11-s + 0.438·12-s + (0.188 − 0.136i)13-s + (0.340 + 1.04i)14-s + (−0.205 + 0.633i)15-s + (1.01 + 0.734i)16-s + (−0.249 − 0.181i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11\)
Sign: $0.886 - 0.462i$
Analytic conductor: \(8.45177\)
Root analytic conductor: \(2.90719\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{11} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 11,\ (\ :11/2),\ 0.886 - 0.462i)\)

Particular Values

\(L(6)\) \(\approx\) \(3.10092 + 0.760270i\)
\(L(\frac12)\) \(\approx\) \(3.10092 + 0.760270i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-2.26e5 + 4.83e5i)T \)
good2 \( 1 + (-44.3 + 32.2i)T + (632. - 1.94e3i)T^{2} \)
3 \( 1 + (-122. - 377. i)T + (-1.43e5 + 1.04e5i)T^{2} \)
5 \( 1 + (-3.99e3 - 2.90e3i)T + (1.50e7 + 4.64e7i)T^{2} \)
7 \( 1 + (1.25e4 - 3.85e4i)T + (-1.59e9 - 1.16e9i)T^{2} \)
13 \( 1 + (-2.52e5 + 1.83e5i)T + (5.53e11 - 1.70e12i)T^{2} \)
17 \( 1 + (1.46e6 + 1.06e6i)T + (1.05e13 + 3.25e13i)T^{2} \)
19 \( 1 + (1.83e6 + 5.65e6i)T + (-9.42e13 + 6.84e13i)T^{2} \)
23 \( 1 + 4.04e7T + 9.52e14T^{2} \)
29 \( 1 + (-3.60e6 + 1.10e7i)T + (-9.87e15 - 7.17e15i)T^{2} \)
31 \( 1 + (2.25e7 - 1.63e7i)T + (7.85e15 - 2.41e16i)T^{2} \)
37 \( 1 + (-1.34e8 + 4.12e8i)T + (-1.43e17 - 1.04e17i)T^{2} \)
41 \( 1 + (1.06e8 + 3.28e8i)T + (-4.45e17 + 3.23e17i)T^{2} \)
43 \( 1 - 1.72e9T + 9.29e17T^{2} \)
47 \( 1 + (8.73e8 + 2.68e9i)T + (-2.00e18 + 1.45e18i)T^{2} \)
53 \( 1 + (-1.06e9 + 7.76e8i)T + (2.86e18 - 8.81e18i)T^{2} \)
59 \( 1 + (2.81e9 - 8.65e9i)T + (-2.43e19 - 1.77e19i)T^{2} \)
61 \( 1 + (-7.15e9 - 5.19e9i)T + (1.34e19 + 4.13e19i)T^{2} \)
67 \( 1 + 1.39e10T + 1.22e20T^{2} \)
71 \( 1 + (2.09e10 + 1.52e10i)T + (7.14e19 + 2.19e20i)T^{2} \)
73 \( 1 + (-3.07e9 + 9.47e9i)T + (-2.53e20 - 1.84e20i)T^{2} \)
79 \( 1 + (3.09e10 - 2.24e10i)T + (2.31e20 - 7.11e20i)T^{2} \)
83 \( 1 + (-3.45e10 - 2.50e10i)T + (3.97e20 + 1.22e21i)T^{2} \)
89 \( 1 - 3.02e10T + 2.77e21T^{2} \)
97 \( 1 + (4.68e10 - 3.40e10i)T + (2.21e21 - 6.80e21i)T^{2} \)
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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

−18.00800427400328502956138370042, −16.10376535833201964301576230190, −14.73164953465349714129814115887, −13.58051656519287661744425732740, −12.02921619677060647226311554224, −10.52591591370962125613612022539, −8.934116712030049067401748388080, −5.82600312680930711368844021217, −3.96769906454105730720405220872, −2.54292601301703375744986298794, 1.46122955220507349215759323091, 4.31094236964406763789635735603, 6.27074151953458215170636179496, 7.52475389499775726505841204006, 9.912011532233955258449980195045, 12.57884506809856709454252051674, 13.44429334264417573076217266790, 14.40199788508519250790607264603, 16.09263499069256109631951399969, 17.47751653654944182574464268580

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