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Properties

Label 2-22-11.10-c8-0-3
Degree $2$
Conductor $22$
Sign $0.980 + 0.195i$
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 − 143.·3-s − 128.·4-s − 309.·5-s − 1.62e3i·6-s + 1.25e3i·7-s − 1.44e3i·8-s + 1.39e4·9-s − 3.50e3i·10-s + (2.85e3 − 1.43e4i)11-s + 1.83e4·12-s + 3.85e4i·13-s − 1.42e4·14-s + 4.44e4·15-s + 1.63e4·16-s − 3.19e4i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.76·3-s − 0.500·4-s − 0.495·5-s − 1.25i·6-s + 0.523i·7-s − 0.353i·8-s + 2.12·9-s − 0.350i·10-s + (0.195 − 0.980i)11-s + 0.884·12-s + 1.35i·13-s − 0.370·14-s + 0.877·15-s + 0.250·16-s − 0.382i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22\)    =    \(2 \cdot 11\)
Sign: $0.980 + 0.195i$
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: \((2,\ 22,\ (\ :4),\ 0.980 + 0.195i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.582923 - 0.0574745i\)
\(L(\frac12)\) \(\approx\) \(0.582923 - 0.0574745i\)
\(L(5)\) 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 \( 1 + (-2.85e3 + 1.43e4i)T \)
good3 \( 1 + 143.T + 6.56e3T^{2} \)
5 \( 1 + 309.T + 3.90e5T^{2} \)
7 \( 1 - 1.25e3iT - 5.76e6T^{2} \)
13 \( 1 - 3.85e4iT - 8.15e8T^{2} \)
17 \( 1 + 3.19e4iT - 6.97e9T^{2} \)
19 \( 1 + 1.96e5iT - 1.69e10T^{2} \)
23 \( 1 - 2.93e5T + 7.83e10T^{2} \)
29 \( 1 + 3.68e5iT - 5.00e11T^{2} \)
31 \( 1 + 4.33e5T + 8.52e11T^{2} \)
37 \( 1 - 2.92e6T + 3.51e12T^{2} \)
41 \( 1 - 3.66e6iT - 7.98e12T^{2} \)
43 \( 1 + 5.40e6iT - 1.16e13T^{2} \)
47 \( 1 - 3.98e6T + 2.38e13T^{2} \)
53 \( 1 - 4.85e6T + 6.22e13T^{2} \)
59 \( 1 - 7.61e5T + 1.46e14T^{2} \)
61 \( 1 + 1.65e7iT - 1.91e14T^{2} \)
67 \( 1 + 3.56e7T + 4.06e14T^{2} \)
71 \( 1 - 6.40e6T + 6.45e14T^{2} \)
73 \( 1 + 1.18e7iT - 8.06e14T^{2} \)
79 \( 1 + 5.34e7iT - 1.51e15T^{2} \)
83 \( 1 + 1.61e7iT - 2.25e15T^{2} \)
89 \( 1 + 1.55e6T + 3.93e15T^{2} \)
97 \( 1 + 1.42e8T + 7.83e15T^{2} \)
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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

−16.32001254743607205648551548314, −15.32388180546903993259545716616, −13.42932235297281762386912470318, −11.85408422479473431947437722372, −11.14499142389372598064129361276, −9.162670140851231802511683781435, −7.07436326359773406932803814775, −5.91633112556146990266727982266, −4.54905278157928811308099504724, −0.48545729405894399113624797199, 1.02698248066841932738401131699, 4.15376144810945625957150341167, 5.65937990792877305618529010626, 7.49086413956371758220498480379, 10.06265930275647607234196107310, 10.90276149331917367583792989581, 12.12940280105085423635631921338, 12.89216540564802160315529820479, 15.07830011305514415146683728931, 16.55257181566500178361867281401

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