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Properties

Label 2-22-11.10-c14-0-3
Degree $2$
Conductor $22$
Sign $0.927 - 0.373i$
Analytic cond. $27.3523$
Root an. cond. $5.22994$
Motivic weight $14$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 90.5i·2-s − 707.·3-s − 8.19e3·4-s − 8.96e3·5-s + 6.40e4i·6-s − 5.31e5i·7-s + 7.41e5i·8-s − 4.28e6·9-s + 8.11e5i·10-s + (−7.28e6 − 1.80e7i)11-s + 5.79e6·12-s + 9.70e7i·13-s − 4.80e7·14-s + 6.34e6·15-s + 6.71e7·16-s − 2.57e8i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.323·3-s − 0.500·4-s − 0.114·5-s + 0.228i·6-s − 0.645i·7-s + 0.353i·8-s − 0.895·9-s + 0.0811i·10-s + (−0.373 − 0.927i)11-s + 0.161·12-s + 1.54i·13-s − 0.456·14-s + 0.0371·15-s + 0.250·16-s − 0.626i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22\)    =    \(2 \cdot 11\)
Sign: $0.927 - 0.373i$
Analytic conductor: \(27.3523\)
Root analytic conductor: \(5.22994\)
Motivic weight: \(14\)
Rational: no
Arithmetic: yes
Character: $\chi_{22} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 22,\ (\ :7),\ 0.927 - 0.373i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(0.9576183851\)
\(L(\frac12)\) \(\approx\) \(0.9576183851\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 90.5iT \)
11 \( 1 + (7.28e6 + 1.80e7i)T \)
good3 \( 1 + 707.T + 4.78e6T^{2} \)
5 \( 1 + 8.96e3T + 6.10e9T^{2} \)
7 \( 1 + 5.31e5iT - 6.78e11T^{2} \)
13 \( 1 - 9.70e7iT - 3.93e15T^{2} \)
17 \( 1 + 2.57e8iT - 1.68e17T^{2} \)
19 \( 1 - 1.10e9iT - 7.99e17T^{2} \)
23 \( 1 - 4.02e9T + 1.15e19T^{2} \)
29 \( 1 + 4.89e9iT - 2.97e20T^{2} \)
31 \( 1 - 4.90e10T + 7.56e20T^{2} \)
37 \( 1 - 1.37e11T + 9.01e21T^{2} \)
41 \( 1 - 3.00e11iT - 3.79e22T^{2} \)
43 \( 1 - 2.97e11iT - 7.38e22T^{2} \)
47 \( 1 - 2.57e11T + 2.56e23T^{2} \)
53 \( 1 + 2.17e12T + 1.37e24T^{2} \)
59 \( 1 + 6.80e11T + 6.19e24T^{2} \)
61 \( 1 + 1.61e12iT - 9.87e24T^{2} \)
67 \( 1 + 1.00e13T + 3.67e25T^{2} \)
71 \( 1 + 4.23e12T + 8.27e25T^{2} \)
73 \( 1 - 2.12e13iT - 1.22e26T^{2} \)
79 \( 1 - 5.80e12iT - 3.68e26T^{2} \)
83 \( 1 - 2.00e13iT - 7.36e26T^{2} \)
89 \( 1 - 5.96e13T + 1.95e27T^{2} \)
97 \( 1 + 1.98e13T + 6.52e27T^{2} \)
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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

−14.34918368469848203585004296353, −13.49157758102242100947983065550, −11.79086845425190552289111811921, −11.08399177613300228812840878311, −9.575860497391805976142385727970, −8.091537602998561360754697123939, −6.16337133122101558674270114884, −4.46043229168831410485211094013, −2.90410251880749723935268862753, −1.03621594426067674437051484649, 0.39870686365676598234023793100, 2.78392170010192233131805544996, 4.92395231955831781938931419017, 6.02240184838369518285566853643, 7.66838061163774947071101025299, 8.939564202939826169782622778979, 10.56572606918502990311274721644, 12.14175038793698004181869203665, 13.34456154380123095600867792619, 15.03124041706493095520848622104

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