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Properties

Label 2-22-11.10-c8-0-0
Degree $2$
Conductor $22$
Sign $0.196 - 0.980i$
Analytic cond. $8.96232$
Root an. cond. $2.99371$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 11.3i·2-s − 1.48·3-s − 128.·4-s − 214.·5-s + 16.7i·6-s + 2.22e3i·7-s + 1.44e3i·8-s − 6.55e3·9-s + 2.42e3i·10-s + (1.43e4 + 2.87e3i)11-s + 190.·12-s + 4.56e4i·13-s + 2.51e4·14-s + 318.·15-s + 1.63e4·16-s + 5.78e4i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.0183·3-s − 0.500·4-s − 0.343·5-s + 0.0129i·6-s + 0.926i·7-s + 0.353i·8-s − 0.999·9-s + 0.242i·10-s + (0.980 + 0.196i)11-s + 0.00916·12-s + 1.59i·13-s + 0.654·14-s + 0.00629·15-s + 0.250·16-s + 0.692i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.196 - 0.980i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.196 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22\)    =    \(2 \cdot 11\)
Sign: $0.196 - 0.980i$
Analytic conductor: \(8.96232\)
Root analytic conductor: \(2.99371\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{22} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 22,\ (\ :4),\ 0.196 - 0.980i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.661906 + 0.542537i\)
\(L(\frac12)\) \(\approx\) \(0.661906 + 0.542537i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 11.3iT \)
11 \( 1 + (-1.43e4 - 2.87e3i)T \)
good3 \( 1 + 1.48T + 6.56e3T^{2} \)
5 \( 1 + 214.T + 3.90e5T^{2} \)
7 \( 1 - 2.22e3iT - 5.76e6T^{2} \)
13 \( 1 - 4.56e4iT - 8.15e8T^{2} \)
17 \( 1 - 5.78e4iT - 6.97e9T^{2} \)
19 \( 1 + 7.00e3iT - 1.69e10T^{2} \)
23 \( 1 + 3.70e5T + 7.83e10T^{2} \)
29 \( 1 + 9.76e5iT - 5.00e11T^{2} \)
31 \( 1 + 9.89e4T + 8.52e11T^{2} \)
37 \( 1 + 1.22e6T + 3.51e12T^{2} \)
41 \( 1 - 3.00e6iT - 7.98e12T^{2} \)
43 \( 1 - 3.17e6iT - 1.16e13T^{2} \)
47 \( 1 + 2.24e6T + 2.38e13T^{2} \)
53 \( 1 - 1.29e7T + 6.22e13T^{2} \)
59 \( 1 - 1.29e7T + 1.46e14T^{2} \)
61 \( 1 + 5.39e6iT - 1.91e14T^{2} \)
67 \( 1 + 1.86e7T + 4.06e14T^{2} \)
71 \( 1 + 5.69e6T + 6.45e14T^{2} \)
73 \( 1 + 1.85e7iT - 8.06e14T^{2} \)
79 \( 1 + 8.41e5iT - 1.51e15T^{2} \)
83 \( 1 - 8.26e7iT - 2.25e15T^{2} \)
89 \( 1 + 1.71e7T + 3.93e15T^{2} \)
97 \( 1 - 1.06e8T + 7.83e15T^{2} \)
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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

−16.56243257399898077361946444461, −14.92823036839404843948655111642, −13.84288353774504249915758380359, −11.97875112455376745247072897851, −11.56232689242975892302635647587, −9.563095257025125051063295117927, −8.424721569542621132479166021896, −6.10294782825630786617269991284, −4.04612678487144315799139279415, −2.06178573603236585289994537476, 0.41823066670150372576833510465, 3.66051839978890102135558141642, 5.62982819876303121011980433642, 7.31026999854019265721562855569, 8.616877224625954603832755496982, 10.36742662129282882691513069793, 11.94233395238123510841109860268, 13.63031618374260446411902420830, 14.57258284347183744326221385717, 15.94527909311567068293043415000

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