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Properties

Label 2-22-11.10-c10-0-9
Degree $2$
Conductor $22$
Sign $-0.663 - 0.747i$
Analytic cond. $13.9778$
Root an. cond. $3.73869$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 22.6i·2-s − 81.3·3-s − 512.·4-s + 2.18e3·5-s + 1.83e3i·6-s − 2.45e4i·7-s + 1.15e4i·8-s − 5.24e4·9-s − 4.95e4i·10-s + (−1.20e5 + 1.06e5i)11-s + 4.16e4·12-s + 1.65e5i·13-s − 5.56e5·14-s − 1.77e5·15-s + 2.62e5·16-s − 2.44e5i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.334·3-s − 0.500·4-s + 0.700·5-s + 0.236i·6-s − 1.46i·7-s + 0.353i·8-s − 0.888·9-s − 0.495i·10-s + (−0.747 + 0.663i)11-s + 0.167·12-s + 0.446i·13-s − 1.03·14-s − 0.234·15-s + 0.250·16-s − 0.172i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.663 - 0.747i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.663 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22\)    =    \(2 \cdot 11\)
Sign: $-0.663 - 0.747i$
Analytic conductor: \(13.9778\)
Root analytic conductor: \(3.73869\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{22} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 22,\ (\ :5),\ -0.663 - 0.747i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.0802443 + 0.178563i\)
\(L(\frac12)\) \(\approx\) \(0.0802443 + 0.178563i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 22.6iT \)
11 \( 1 + (1.20e5 - 1.06e5i)T \)
good3 \( 1 + 81.3T + 5.90e4T^{2} \)
5 \( 1 - 2.18e3T + 9.76e6T^{2} \)
7 \( 1 + 2.45e4iT - 2.82e8T^{2} \)
13 \( 1 - 1.65e5iT - 1.37e11T^{2} \)
17 \( 1 + 2.44e5iT - 2.01e12T^{2} \)
19 \( 1 - 3.94e6iT - 6.13e12T^{2} \)
23 \( 1 + 1.16e7T + 4.14e13T^{2} \)
29 \( 1 + 1.49e7iT - 4.20e14T^{2} \)
31 \( 1 + 4.11e7T + 8.19e14T^{2} \)
37 \( 1 - 8.07e7T + 4.80e15T^{2} \)
41 \( 1 - 5.19e7iT - 1.34e16T^{2} \)
43 \( 1 + 2.77e8iT - 2.16e16T^{2} \)
47 \( 1 - 1.23e8T + 5.25e16T^{2} \)
53 \( 1 - 5.22e7T + 1.74e17T^{2} \)
59 \( 1 + 9.40e8T + 5.11e17T^{2} \)
61 \( 1 - 2.99e8iT - 7.13e17T^{2} \)
67 \( 1 - 1.61e9T + 1.82e18T^{2} \)
71 \( 1 + 7.90e8T + 3.25e18T^{2} \)
73 \( 1 + 1.36e9iT - 4.29e18T^{2} \)
79 \( 1 + 1.52e9iT - 9.46e18T^{2} \)
83 \( 1 + 5.40e9iT - 1.55e19T^{2} \)
89 \( 1 + 2.91e9T + 3.11e19T^{2} \)
97 \( 1 + 7.67e9T + 7.37e19T^{2} \)
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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

−14.32645278906853238569499043911, −13.54001294071703136619262588857, −12.07566143358459968660083106143, −10.63970559067554586536976015503, −9.791671503798882348813834075449, −7.78764860696902791109714694856, −5.80383468904273677584534468731, −4.00897497038895651444915411344, −1.93596469503109269348027762775, −0.07866926166237412298020858725, 2.59785374822927017213180202839, 5.38802565148475463958502108482, 6.04626872378388444142336315530, 8.218282084679831586243610656748, 9.371808480184224106135555448973, 11.18120428583327532623323200350, 12.70633799997599169968528966696, 14.01491311104049406001336555968, 15.29924127025918304224696623855, 16.33370259566991364761365993791

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