Properties

Label 2-11-11.9-c9-0-5
Degree $2$
Conductor $11$
Sign $0.626 + 0.779i$
Analytic cond. $5.66539$
Root an. cond. $2.38020$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−13.1 + 40.5i)2-s + (121. − 88.5i)3-s + (−1.05e3 − 767. i)4-s + (−602. − 1.85e3i)5-s + (1.98e3 + 6.10e3i)6-s + (−3.57e3 − 2.59e3i)7-s + (2.73e4 − 1.98e4i)8-s + (923. − 2.84e3i)9-s + 8.31e4·10-s + (−4.85e4 + 157. i)11-s − 1.96e5·12-s + (−8.75e3 + 2.69e4i)13-s + (1.52e5 − 1.10e5i)14-s + (−2.37e5 − 1.72e5i)15-s + (2.38e5 + 7.35e5i)16-s + (−9.51e4 − 2.92e5i)17-s + ⋯
L(s)  = 1  + (−0.582 + 1.79i)2-s + (0.868 − 0.630i)3-s + (−2.06 − 1.49i)4-s + (−0.431 − 1.32i)5-s + (0.624 + 1.92i)6-s + (−0.562 − 0.409i)7-s + (2.36 − 1.71i)8-s + (0.0469 − 0.144i)9-s + 2.62·10-s + (−0.999 + 0.00323i)11-s − 2.73·12-s + (−0.0850 + 0.261i)13-s + (1.06 − 0.770i)14-s + (−1.21 − 0.880i)15-s + (0.911 + 2.80i)16-s + (−0.276 − 0.850i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11\)
Sign: $0.626 + 0.779i$
Analytic conductor: \(5.66539\)
Root analytic conductor: \(2.38020\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{11} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 11,\ (\ :9/2),\ 0.626 + 0.779i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.674228 - 0.322984i\)
\(L(\frac12)\) \(\approx\) \(0.674228 - 0.322984i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (4.85e4 - 157. i)T \)
good2 \( 1 + (13.1 - 40.5i)T + (-414. - 300. i)T^{2} \)
3 \( 1 + (-121. + 88.5i)T + (6.08e3 - 1.87e4i)T^{2} \)
5 \( 1 + (602. + 1.85e3i)T + (-1.58e6 + 1.14e6i)T^{2} \)
7 \( 1 + (3.57e3 + 2.59e3i)T + (1.24e7 + 3.83e7i)T^{2} \)
13 \( 1 + (8.75e3 - 2.69e4i)T + (-8.57e9 - 6.23e9i)T^{2} \)
17 \( 1 + (9.51e4 + 2.92e5i)T + (-9.59e10 + 6.97e10i)T^{2} \)
19 \( 1 + (-6.19e5 + 4.49e5i)T + (9.97e10 - 3.06e11i)T^{2} \)
23 \( 1 + 6.97e5T + 1.80e12T^{2} \)
29 \( 1 + (-1.46e6 - 1.06e6i)T + (4.48e12 + 1.37e13i)T^{2} \)
31 \( 1 + (-1.26e6 + 3.89e6i)T + (-2.13e13 - 1.55e13i)T^{2} \)
37 \( 1 + (9.42e6 + 6.84e6i)T + (4.01e13 + 1.23e14i)T^{2} \)
41 \( 1 + (3.45e6 - 2.50e6i)T + (1.01e14 - 3.11e14i)T^{2} \)
43 \( 1 - 2.59e7T + 5.02e14T^{2} \)
47 \( 1 + (2.88e7 - 2.09e7i)T + (3.45e14 - 1.06e15i)T^{2} \)
53 \( 1 + (-1.82e7 + 5.60e7i)T + (-2.66e15 - 1.93e15i)T^{2} \)
59 \( 1 + (3.34e7 + 2.42e7i)T + (2.67e15 + 8.23e15i)T^{2} \)
61 \( 1 + (-8.83e6 - 2.71e7i)T + (-9.46e15 + 6.87e15i)T^{2} \)
67 \( 1 - 1.49e8T + 2.72e16T^{2} \)
71 \( 1 + (3.71e7 + 1.14e8i)T + (-3.70e16 + 2.69e16i)T^{2} \)
73 \( 1 + (-2.84e8 - 2.07e8i)T + (1.81e16 + 5.59e16i)T^{2} \)
79 \( 1 + (-1.20e8 + 3.70e8i)T + (-9.69e16 - 7.04e16i)T^{2} \)
83 \( 1 + (1.12e8 + 3.46e8i)T + (-1.51e17 + 1.09e17i)T^{2} \)
89 \( 1 + 1.44e8T + 3.50e17T^{2} \)
97 \( 1 + (-2.22e8 + 6.84e8i)T + (-6.15e17 - 4.46e17i)T^{2} \)
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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

−17.91946068769058359374616457620, −16.40499212237882403229158989410, −15.76080092569110422904935586576, −13.94750810856926328778936656648, −13.07751948139968851819883592180, −9.430431992478210253971340053982, −8.265263797052846962231382125585, −7.25268431702043625551799200561, −5.02433789045258011732782705022, −0.48683049315226006842212419010, 2.68918855359055162523342575940, 3.56963096263225865742174700880, 8.180642275629588970916981806046, 9.786051896271674385867379400247, 10.67046039380726867975443719596, 12.25553947617372388846296845538, 13.96931846464431476922886594492, 15.52714172278313739750905070328, 17.98057660113611988880232492767, 18.91626186472218550006069247511

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