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Properties

Label 2-11-11.10-c12-0-2
Degree $2$
Conductor $11$
Sign $-0.669 - 0.743i$
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  + 45.2i·2-s − 583.·3-s + 2.04e3·4-s + 1.58e4·5-s − 2.64e4i·6-s + 9.38e4i·7-s + 2.78e5i·8-s − 1.90e5·9-s + 7.19e5i·10-s + (−1.18e6 − 1.31e6i)11-s − 1.19e6·12-s + 5.35e6i·13-s − 4.24e6·14-s − 9.27e6·15-s − 4.19e6·16-s + 4.63e7i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.800·3-s + 0.499·4-s + 1.01·5-s − 0.566i·6-s + 0.797i·7-s + 1.06i·8-s − 0.358·9-s + 0.719i·10-s + (−0.669 − 0.743i)11-s − 0.400·12-s + 1.10i·13-s − 0.564·14-s − 0.814·15-s − 0.250·16-s + 1.92i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.669 - 0.743i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.669 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11\)
Sign: $-0.669 - 0.743i$
Analytic conductor: \(10.0539\)
Root analytic conductor: \(3.17079\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{11} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 11,\ (\ :6),\ -0.669 - 0.743i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.603047 + 1.35482i\)
\(L(\frac12)\) \(\approx\) \(0.603047 + 1.35482i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (1.18e6 + 1.31e6i)T \)
good2 \( 1 - 45.2iT - 4.09e3T^{2} \)
3 \( 1 + 583.T + 5.31e5T^{2} \)
5 \( 1 - 1.58e4T + 2.44e8T^{2} \)
7 \( 1 - 9.38e4iT - 1.38e10T^{2} \)
13 \( 1 - 5.35e6iT - 2.32e13T^{2} \)
17 \( 1 - 4.63e7iT - 5.82e14T^{2} \)
19 \( 1 - 4.52e6iT - 2.21e15T^{2} \)
23 \( 1 - 8.67e4T + 2.19e16T^{2} \)
29 \( 1 + 5.83e8iT - 3.53e17T^{2} \)
31 \( 1 - 7.40e8T + 7.87e17T^{2} \)
37 \( 1 + 3.08e9T + 6.58e18T^{2} \)
41 \( 1 - 1.18e9iT - 2.25e19T^{2} \)
43 \( 1 + 3.38e9iT - 3.99e19T^{2} \)
47 \( 1 - 1.92e10T + 1.16e20T^{2} \)
53 \( 1 - 9.28e8T + 4.91e20T^{2} \)
59 \( 1 - 1.61e10T + 1.77e21T^{2} \)
61 \( 1 - 8.95e9iT - 2.65e21T^{2} \)
67 \( 1 - 1.68e10T + 8.18e21T^{2} \)
71 \( 1 - 1.56e11T + 1.64e22T^{2} \)
73 \( 1 + 2.47e11iT - 2.29e22T^{2} \)
79 \( 1 - 1.69e11iT - 5.90e22T^{2} \)
83 \( 1 + 4.27e11iT - 1.06e23T^{2} \)
89 \( 1 - 2.30e11T + 2.46e23T^{2} \)
97 \( 1 + 1.43e12T + 6.93e23T^{2} \)
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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

−17.49902294590056106973605708850, −16.77170842859850807210016325603, −15.41120364497648462828679011668, −13.91688239636464942981355553373, −12.00126179161512720127291933186, −10.64689714573956019765854644949, −8.531752775350340024868969400604, −6.26386777535615118820445495052, −5.62173049209927548083272526846, −2.14189057731434925897486088553, 0.74077947015724377286071077394, 2.66094414196427942264989929428, 5.34489695762179773341783113572, 7.06422982138893178711259535480, 9.932759548687673264635875867694, 10.86674677103688777933790224138, 12.32097981112176794276952801326, 13.74175717559784227451506842687, 15.81236288953762488456344632410, 17.21230749681682211737576018482

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