Properties

Label 2-11-11.6-c12-0-5
Degree $2$
Conductor $11$
Sign $0.916 + 0.400i$
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  + (−8.21 + 2.66i)2-s + (−83.6 − 60.8i)3-s + (−3.25e3 + 2.36e3i)4-s + (3.57e3 − 1.10e4i)5-s + (849. + 276. i)6-s + (6.12e4 + 8.42e4i)7-s + (4.12e4 − 5.67e4i)8-s + (−1.60e5 − 4.95e5i)9-s + 1.00e5i·10-s + (1.75e6 + 2.11e5i)11-s + 4.16e5·12-s + (8.42e6 − 2.73e6i)13-s + (−7.27e5 − 5.28e5i)14-s + (−9.69e5 + 7.04e5i)15-s + (4.90e6 − 1.50e7i)16-s + (1.10e7 + 3.58e6i)17-s + ⋯
L(s)  = 1  + (−0.128 + 0.0416i)2-s + (−0.114 − 0.0834i)3-s + (−0.794 + 0.577i)4-s + (0.229 − 0.705i)5-s + (0.0182 + 0.00591i)6-s + (0.520 + 0.716i)7-s + (0.157 − 0.216i)8-s + (−0.302 − 0.931i)9-s + 0.100i·10-s + (0.992 + 0.119i)11-s + 0.139·12-s + (1.74 − 0.566i)13-s + (−0.0966 − 0.0702i)14-s + (−0.0851 + 0.0618i)15-s + (0.292 − 0.899i)16-s + (0.457 + 0.148i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11\)
Sign: $0.916 + 0.400i$
Analytic conductor: \(10.0539\)
Root analytic conductor: \(3.17079\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{11} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 11,\ (\ :6),\ 0.916 + 0.400i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(1.46363 - 0.305779i\)
\(L(\frac12)\) \(\approx\) \(1.46363 - 0.305779i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-1.75e6 - 2.11e5i)T \)
good2 \( 1 + (8.21 - 2.66i)T + (3.31e3 - 2.40e3i)T^{2} \)
3 \( 1 + (83.6 + 60.8i)T + (1.64e5 + 5.05e5i)T^{2} \)
5 \( 1 + (-3.57e3 + 1.10e4i)T + (-1.97e8 - 1.43e8i)T^{2} \)
7 \( 1 + (-6.12e4 - 8.42e4i)T + (-4.27e9 + 1.31e10i)T^{2} \)
13 \( 1 + (-8.42e6 + 2.73e6i)T + (1.88e13 - 1.36e13i)T^{2} \)
17 \( 1 + (-1.10e7 - 3.58e6i)T + (4.71e14 + 3.42e14i)T^{2} \)
19 \( 1 + (-1.20e7 + 1.65e7i)T + (-6.83e14 - 2.10e15i)T^{2} \)
23 \( 1 + 2.79e8T + 2.19e16T^{2} \)
29 \( 1 + (-3.86e8 - 5.31e8i)T + (-1.09e17 + 3.36e17i)T^{2} \)
31 \( 1 + (3.45e8 + 1.06e9i)T + (-6.37e17 + 4.62e17i)T^{2} \)
37 \( 1 + (-2.63e9 + 1.91e9i)T + (2.03e18 - 6.26e18i)T^{2} \)
41 \( 1 + (-3.21e9 + 4.42e9i)T + (-6.97e18 - 2.14e19i)T^{2} \)
43 \( 1 - 6.76e9iT - 3.99e19T^{2} \)
47 \( 1 + (3.26e9 + 2.37e9i)T + (3.59e19 + 1.10e20i)T^{2} \)
53 \( 1 + (1.41e9 + 4.34e9i)T + (-3.97e20 + 2.88e20i)T^{2} \)
59 \( 1 + (-9.97e9 + 7.24e9i)T + (5.49e20 - 1.69e21i)T^{2} \)
61 \( 1 + (-4.88e10 - 1.58e10i)T + (2.14e21 + 1.56e21i)T^{2} \)
67 \( 1 - 2.92e10T + 8.18e21T^{2} \)
71 \( 1 + (2.97e10 - 9.16e10i)T + (-1.32e22 - 9.64e21i)T^{2} \)
73 \( 1 + (1.00e11 + 1.37e11i)T + (-7.07e21 + 2.17e22i)T^{2} \)
79 \( 1 + (-4.43e9 + 1.44e9i)T + (4.78e22 - 3.47e22i)T^{2} \)
83 \( 1 + (1.76e11 + 5.71e10i)T + (8.64e22 + 6.28e22i)T^{2} \)
89 \( 1 + 3.19e11T + 2.46e23T^{2} \)
97 \( 1 + (-1.79e11 - 5.52e11i)T + (-5.61e23 + 4.07e23i)T^{2} \)
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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

−17.65125543313224909070452293788, −16.22813433036136701226815445199, −14.43222445710013488902620050281, −12.91125998383776489905071320770, −11.74526921509630750978111450875, −9.278695215813898757928228751652, −8.356774098993102526736747648513, −5.81605059900089701808951715763, −3.84549030480720994157657063525, −0.980318943634965984165218442312, 1.33861660699494774798075760044, 4.14638887073024619438924168621, 6.10976661430958580373424058425, 8.318090262943481674805549900096, 10.13265318961408168798855332830, 11.26100033992194101579484974325, 13.86026820973248466534675243379, 14.20343127192212222163893539979, 16.34070460341180869195625366235, 17.79637937237061865924101642444

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