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Properties

Label 2-22-11.10-c6-0-3
Degree $2$
Conductor $22$
Sign $0.471 + 0.881i$
Analytic cond. $5.06118$
Root an. cond. $2.24970$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.65i·2-s − 41.5·3-s − 32.0·4-s + 155.·5-s − 234. i·6-s − 562. i·7-s − 181. i·8-s + 994.·9-s + 880. i·10-s + (−1.17e3 + 627. i)11-s + 1.32e3·12-s − 3.27e3i·13-s + 3.18e3·14-s − 6.45e3·15-s + 1.02e3·16-s + 1.75e3i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.53·3-s − 0.500·4-s + 1.24·5-s − 1.08i·6-s − 1.64i·7-s − 0.353i·8-s + 1.36·9-s + 0.880i·10-s + (−0.881 + 0.471i)11-s + 0.768·12-s − 1.48i·13-s + 1.15·14-s − 1.91·15-s + 0.250·16-s + 0.356i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 + 0.881i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.471 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22\)    =    \(2 \cdot 11\)
Sign: $0.471 + 0.881i$
Analytic conductor: \(5.06118\)
Root analytic conductor: \(2.24970\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{22} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 22,\ (\ :3),\ 0.471 + 0.881i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.653118 - 0.391290i\)
\(L(\frac12)\) \(\approx\) \(0.653118 - 0.391290i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 5.65iT \)
11 \( 1 + (1.17e3 - 627. i)T \)
good3 \( 1 + 41.5T + 729T^{2} \)
5 \( 1 - 155.T + 1.56e4T^{2} \)
7 \( 1 + 562. iT - 1.17e5T^{2} \)
13 \( 1 + 3.27e3iT - 4.82e6T^{2} \)
17 \( 1 - 1.75e3iT - 2.41e7T^{2} \)
19 \( 1 + 8.80e3iT - 4.70e7T^{2} \)
23 \( 1 + 4.91e3T + 1.48e8T^{2} \)
29 \( 1 + 1.50e4iT - 5.94e8T^{2} \)
31 \( 1 + 4.42e3T + 8.87e8T^{2} \)
37 \( 1 + 2.73e4T + 2.56e9T^{2} \)
41 \( 1 - 4.01e4iT - 4.75e9T^{2} \)
43 \( 1 - 1.33e4iT - 6.32e9T^{2} \)
47 \( 1 - 1.25e5T + 1.07e10T^{2} \)
53 \( 1 - 1.00e5T + 2.21e10T^{2} \)
59 \( 1 + 3.37e5T + 4.21e10T^{2} \)
61 \( 1 - 5.37e4iT - 5.15e10T^{2} \)
67 \( 1 - 1.53e5T + 9.04e10T^{2} \)
71 \( 1 - 1.18e5T + 1.28e11T^{2} \)
73 \( 1 + 4.02e5iT - 1.51e11T^{2} \)
79 \( 1 - 5.61e5iT - 2.43e11T^{2} \)
83 \( 1 + 5.69e5iT - 3.26e11T^{2} \)
89 \( 1 - 8.05e5T + 4.96e11T^{2} \)
97 \( 1 - 1.00e6T + 8.32e11T^{2} \)
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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

−16.87756578969324380906439875950, −15.52744420141141511090741596286, −13.66611407906493487371984354918, −12.86419973777222459091895832968, −10.70692858944678171007378567512, −10.07035550979779544212167280753, −7.42897562937392150075326987693, −6.07481631404795758352013547369, −4.89679861497867780779587252285, −0.56662782128803702968651739375, 1.99710007934803792864318452153, 5.27850361108800823546382264666, 6.08100850016220483011935435721, 9.089054287754609490665832589148, 10.38512039758837976905650411567, 11.69769410659396590923392380463, 12.51814540227003293421702507368, 14.03278412958385377314814679942, 15.99636877058941203616520838521, 17.11470705395974032847529561543

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