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Properties

Label 2-22-11.10-c8-0-4
Degree $2$
Conductor $22$
Sign $0.820 + 0.571i$
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 + 128.·3-s − 128.·4-s + 885.·5-s − 1.45e3i·6-s + 2.93e3i·7-s + 1.44e3i·8-s + 9.92e3·9-s − 1.00e4i·10-s + (−8.36e3 + 1.20e4i)11-s − 1.64e4·12-s − 4.71e4i·13-s + 3.32e4·14-s + 1.13e5·15-s + 1.63e4·16-s − 1.17e5i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.58·3-s − 0.500·4-s + 1.41·5-s − 1.12i·6-s + 1.22i·7-s + 0.353i·8-s + 1.51·9-s − 1.00i·10-s + (−0.571 + 0.820i)11-s − 0.792·12-s − 1.65i·13-s + 0.864·14-s + 2.24·15-s + 0.250·16-s − 1.41i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22\)    =    \(2 \cdot 11\)
Sign: $0.820 + 0.571i$
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: \((2,\ 22,\ (\ :4),\ 0.820 + 0.571i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.93527 - 0.921379i\)
\(L(\frac12)\) \(\approx\) \(2.93527 - 0.921379i\)
\(L(5)\) 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 \( 1 + (8.36e3 - 1.20e4i)T \)
good3 \( 1 - 128.T + 6.56e3T^{2} \)
5 \( 1 - 885.T + 3.90e5T^{2} \)
7 \( 1 - 2.93e3iT - 5.76e6T^{2} \)
13 \( 1 + 4.71e4iT - 8.15e8T^{2} \)
17 \( 1 + 1.17e5iT - 6.97e9T^{2} \)
19 \( 1 - 4.08e4iT - 1.69e10T^{2} \)
23 \( 1 + 4.53e5T + 7.83e10T^{2} \)
29 \( 1 - 3.85e5iT - 5.00e11T^{2} \)
31 \( 1 + 1.59e5T + 8.52e11T^{2} \)
37 \( 1 - 7.56e5T + 3.51e12T^{2} \)
41 \( 1 - 8.50e5iT - 7.98e12T^{2} \)
43 \( 1 + 5.07e5iT - 1.16e13T^{2} \)
47 \( 1 + 2.44e6T + 2.38e13T^{2} \)
53 \( 1 + 1.92e5T + 6.22e13T^{2} \)
59 \( 1 + 3.19e5T + 1.46e14T^{2} \)
61 \( 1 + 1.49e6iT - 1.91e14T^{2} \)
67 \( 1 + 1.51e7T + 4.06e14T^{2} \)
71 \( 1 - 4.30e7T + 6.45e14T^{2} \)
73 \( 1 - 1.16e7iT - 8.06e14T^{2} \)
79 \( 1 - 4.23e7iT - 1.51e15T^{2} \)
83 \( 1 - 1.93e7iT - 2.25e15T^{2} \)
89 \( 1 - 8.91e7T + 3.93e15T^{2} \)
97 \( 1 - 4.50e7T + 7.83e15T^{2} \)
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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

−15.58405896527527678065706769954, −14.46685078404578087724786916444, −13.43912747677860631037244234274, −12.47096482602120410211771589535, −10.07263532143503041957729226842, −9.371543801309040370915389333234, −8.048774713233401101749213466025, −5.42084560798211768761279874548, −2.84942332803372134867306334795, −2.06732096785603401930655542710, 1.89986061144622547544825804752, 3.98093630823275878176076431412, 6.33814153297665614653584668815, 7.936842624110965287130673319265, 9.193259798288353233930179939302, 10.27985944652554467191287431918, 13.35069189629619015781145308329, 13.80300430559590288069408026826, 14.56437223019605476748546049667, 16.25545334009673275713098753162

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