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Properties

Label 2-22-11.9-c1-0-0
Degree $2$
Conductor $22$
Sign $0.944 + 0.329i$
Analytic cond. $0.175670$
Root an. cond. $0.419131$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 − 0.951i)2-s + (−2.11 + 1.53i)3-s + (−0.809 − 0.587i)4-s + (−0.381 − 1.17i)5-s + (0.809 + 2.48i)6-s + (1.61 + 1.17i)7-s + (−0.809 + 0.587i)8-s + (1.19 − 3.66i)9-s − 1.23·10-s + (−0.809 − 3.21i)11-s + 2.61·12-s + (−1 + 3.07i)13-s + (1.61 − 1.17i)14-s + (2.61 + 1.90i)15-s + (0.309 + 0.951i)16-s + (0.5 + 1.53i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−1.22 + 0.888i)3-s + (−0.404 − 0.293i)4-s + (−0.170 − 0.525i)5-s + (0.330 + 1.01i)6-s + (0.611 + 0.444i)7-s + (−0.286 + 0.207i)8-s + (0.396 − 1.22i)9-s − 0.390·10-s + (−0.243 − 0.969i)11-s + 0.755·12-s + (−0.277 + 0.853i)13-s + (0.432 − 0.314i)14-s + (0.675 + 0.491i)15-s + (0.0772 + 0.237i)16-s + (0.121 + 0.373i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.329i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22\)    =    \(2 \cdot 11\)
Sign: $0.944 + 0.329i$
Analytic conductor: \(0.175670\)
Root analytic conductor: \(0.419131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{22} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 22,\ (\ :1/2),\ 0.944 + 0.329i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.511272 - 0.0867492i\)
\(L(\frac12)\) \(\approx\) \(0.511272 - 0.0867492i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (0.809 + 3.21i)T \)
good3 \( 1 + (2.11 - 1.53i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (0.381 + 1.17i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-1.61 - 1.17i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (1 - 3.07i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.5 - 1.53i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.690 - 0.502i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 3.23T + 23T^{2} \)
29 \( 1 + (-3.61 - 2.62i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.618 + 1.90i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (7.85 + 5.70i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (2.73 - 1.98i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 11.5T + 43T^{2} \)
47 \( 1 + (2 - 1.45i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.23 - 9.95i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-5.16 - 3.75i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2 - 6.15i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 0.0901T + 67T^{2} \)
71 \( 1 + (0.236 + 0.726i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (10.2 + 7.41i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (4.14 - 12.7i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.95 + 6.01i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 3.09T + 89T^{2} \)
97 \( 1 + (-4.28 + 13.1i)T + (-78.4 - 57.0i)T^{2} \)
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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

−17.95483094654327296346104780977, −16.75267934589628297956076390330, −15.81556046515541937865318033726, −14.22311750941515217791374070327, −12.36241295443237250859504660591, −11.44619607659213608835038775453, −10.40424140172056982202672856051, −8.768998469539068183496117711620, −5.74400588385165905372239236141, −4.44474181454379931595270600981, 5.09333879761163886424786304608, 6.74059272803428382070771594247, 7.75096726427605455178351016739, 10.43192667523290300547890412354, 11.84533436676177067387993792360, 12.94620107202840274194465400940, 14.42000970686806351459148093577, 15.77095734685087974207176901250, 17.38499608753431154795852599274, 17.69551489599528086192959449819

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