Properties

Label 2-11-11.7-c12-0-1
Degree $2$
Conductor $11$
Sign $-0.335 - 0.941i$
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  + (68.6 − 94.5i)2-s + (−308. + 950. i)3-s + (−2.95e3 − 9.08e3i)4-s + (−1.01e4 + 7.39e3i)5-s + (6.86e4 + 9.44e4i)6-s + (−1.64e5 + 5.33e4i)7-s + (−6.06e5 − 1.96e5i)8-s + (−3.78e5 − 2.74e5i)9-s + 1.46e6i·10-s + (−8.96e5 + 1.52e6i)11-s + 9.54e6·12-s + (3.89e6 − 5.36e6i)13-s + (−6.22e6 + 1.91e7i)14-s + (−3.88e6 − 1.19e7i)15-s + (−2.85e7 + 2.07e7i)16-s + (−3.43e6 − 4.73e6i)17-s + ⋯
L(s)  = 1  + (1.07 − 1.47i)2-s + (−0.423 + 1.30i)3-s + (−0.720 − 2.21i)4-s + (−0.651 + 0.473i)5-s + (1.47 + 2.02i)6-s + (−1.39 + 0.453i)7-s + (−2.31 − 0.751i)8-s + (−0.711 − 0.516i)9-s + 1.46i·10-s + (−0.506 + 0.862i)11-s + 3.19·12-s + (0.807 − 1.11i)13-s + (−0.827 + 2.54i)14-s + (−0.340 − 1.04i)15-s + (−1.70 + 1.23i)16-s + (−0.142 − 0.196i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.335 - 0.941i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.335 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11\)
Sign: $-0.335 - 0.941i$
Analytic conductor: \(10.0539\)
Root analytic conductor: \(3.17079\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{11} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 11,\ (\ :6),\ -0.335 - 0.941i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.224803 + 0.318747i\)
\(L(\frac12)\) \(\approx\) \(0.224803 + 0.318747i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (8.96e5 - 1.52e6i)T \)
good2 \( 1 + (-68.6 + 94.5i)T + (-1.26e3 - 3.89e3i)T^{2} \)
3 \( 1 + (308. - 950. i)T + (-4.29e5 - 3.12e5i)T^{2} \)
5 \( 1 + (1.01e4 - 7.39e3i)T + (7.54e7 - 2.32e8i)T^{2} \)
7 \( 1 + (1.64e5 - 5.33e4i)T + (1.11e10 - 8.13e9i)T^{2} \)
13 \( 1 + (-3.89e6 + 5.36e6i)T + (-7.19e12 - 2.21e13i)T^{2} \)
17 \( 1 + (3.43e6 + 4.73e6i)T + (-1.80e14 + 5.54e14i)T^{2} \)
19 \( 1 + (-5.99e6 - 1.94e6i)T + (1.79e15 + 1.30e15i)T^{2} \)
23 \( 1 + 2.11e8T + 2.19e16T^{2} \)
29 \( 1 + (-4.72e8 + 1.53e8i)T + (2.86e17 - 2.07e17i)T^{2} \)
31 \( 1 + (5.46e8 + 3.97e8i)T + (2.43e17 + 7.49e17i)T^{2} \)
37 \( 1 + (-8.54e8 - 2.62e9i)T + (-5.32e18 + 3.86e18i)T^{2} \)
41 \( 1 + (3.25e9 + 1.05e9i)T + (1.82e19 + 1.32e19i)T^{2} \)
43 \( 1 + 3.05e9iT - 3.99e19T^{2} \)
47 \( 1 + (2.95e9 - 9.09e9i)T + (-9.40e19 - 6.82e19i)T^{2} \)
53 \( 1 + (-2.70e10 - 1.96e10i)T + (1.51e20 + 4.67e20i)T^{2} \)
59 \( 1 + (1.38e9 + 4.26e9i)T + (-1.43e21 + 1.04e21i)T^{2} \)
61 \( 1 + (-2.91e10 - 4.00e10i)T + (-8.20e20 + 2.52e21i)T^{2} \)
67 \( 1 + 8.89e10T + 8.18e21T^{2} \)
71 \( 1 + (7.58e10 - 5.50e10i)T + (5.07e21 - 1.56e22i)T^{2} \)
73 \( 1 + (-1.00e11 + 3.27e10i)T + (1.85e22 - 1.34e22i)T^{2} \)
79 \( 1 + (1.59e11 - 2.19e11i)T + (-1.82e22 - 5.61e22i)T^{2} \)
83 \( 1 + (1.56e11 + 2.15e11i)T + (-3.30e22 + 1.01e23i)T^{2} \)
89 \( 1 - 8.95e10T + 2.46e23T^{2} \)
97 \( 1 + (7.56e11 + 5.49e11i)T + (2.14e23 + 6.59e23i)T^{2} \)
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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

−18.46565735754265416796531741777, −15.82084499246641814755811504119, −15.18088709144814439727398814985, −13.19504578360925671241003957506, −11.91521949222050679146238843981, −10.54246494313787167757536346521, −9.790788048661550602078956997980, −5.69062979155489405116188852349, −4.06162164548283932966299014812, −2.98166729792639938808236883365, 0.13163029758020456886115463537, 3.82337679248026658417396526690, 6.02540791360936481620387668559, 6.91661701415909176098808547187, 8.309698783706632588127537888948, 12.01882999291731866114910195386, 13.08143171758552601273987080753, 13.84721596067746162328777765866, 16.06359113340435911363484050645, 16.41042234240895994027943164089

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