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Properties

Label 2-22-11.10-c10-0-1
Degree $2$
Conductor $22$
Sign $0.0252 - 0.999i$
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 + 189.·3-s − 512.·4-s − 1.49e3·5-s − 4.27e3i·6-s + 2.18e4i·7-s + 1.15e4i·8-s − 2.32e4·9-s + 3.37e4i·10-s + (−1.60e5 − 4.06e3i)11-s − 9.68e4·12-s − 4.98e4i·13-s + 4.95e5·14-s − 2.82e5·15-s + 2.62e5·16-s + 2.49e6i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.778·3-s − 0.500·4-s − 0.477·5-s − 0.550i·6-s + 1.30i·7-s + 0.353i·8-s − 0.394·9-s + 0.337i·10-s + (−0.999 − 0.0252i)11-s − 0.389·12-s − 0.134i·13-s + 0.921·14-s − 0.371·15-s + 0.250·16-s + 1.75i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0252 - 0.999i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.0252 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22\)    =    \(2 \cdot 11\)
Sign: $0.0252 - 0.999i$
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: \((2,\ 22,\ (\ :5),\ 0.0252 - 0.999i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.714941 + 0.697106i\)
\(L(\frac12)\) \(\approx\) \(0.714941 + 0.697106i\)
\(L(6)\) 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 \( 1 + (1.60e5 + 4.06e3i)T \)
good3 \( 1 - 189.T + 5.90e4T^{2} \)
5 \( 1 + 1.49e3T + 9.76e6T^{2} \)
7 \( 1 - 2.18e4iT - 2.82e8T^{2} \)
13 \( 1 + 4.98e4iT - 1.37e11T^{2} \)
17 \( 1 - 2.49e6iT - 2.01e12T^{2} \)
19 \( 1 - 1.02e6iT - 6.13e12T^{2} \)
23 \( 1 - 9.76e6T + 4.14e13T^{2} \)
29 \( 1 - 6.29e6iT - 4.20e14T^{2} \)
31 \( 1 - 3.66e6T + 8.19e14T^{2} \)
37 \( 1 + 6.77e7T + 4.80e15T^{2} \)
41 \( 1 + 1.77e8iT - 1.34e16T^{2} \)
43 \( 1 + 4.27e7iT - 2.16e16T^{2} \)
47 \( 1 + 1.23e8T + 5.25e16T^{2} \)
53 \( 1 + 4.18e8T + 1.74e17T^{2} \)
59 \( 1 - 5.37e8T + 5.11e17T^{2} \)
61 \( 1 - 9.73e8iT - 7.13e17T^{2} \)
67 \( 1 + 1.10e9T + 1.82e18T^{2} \)
71 \( 1 - 2.08e9T + 3.25e18T^{2} \)
73 \( 1 - 1.22e9iT - 4.29e18T^{2} \)
79 \( 1 + 4.55e9iT - 9.46e18T^{2} \)
83 \( 1 - 3.69e9iT - 1.55e19T^{2} \)
89 \( 1 - 1.01e10T + 3.11e19T^{2} \)
97 \( 1 + 3.66e9T + 7.37e19T^{2} \)
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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

−15.54239690505110788441097116449, −14.72526502319181667661600765789, −13.15815501380476827410062863463, −12.08118272748131281130373406980, −10.65700055751669160811438298450, −8.955700645931171207003657861049, −8.104197957947058153445065259479, −5.51694347177096850227396429012, −3.41688463706375517013764863627, −2.14806899682348860208750136842, 0.36092918562689343749649907528, 3.13315121202650833558531904784, 4.84801873287245121481283690498, 7.11010564535613675848888381801, 8.045462992711586902471717710465, 9.551823044126755109087743497488, 11.21648322784322139040757537519, 13.29147270017319521310734144032, 14.03303644895128190970632555880, 15.30985938886325716444136148392

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