Properties

Label 2-22-11.3-c1-0-0
Degree $2$
Conductor $22$
Sign $0.927 - 0.374i$
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  + (−0.809 + 0.587i)2-s + (0.118 + 0.363i)3-s + (0.309 − 0.951i)4-s + (−2.61 − 1.90i)5-s + (−0.309 − 0.224i)6-s + (−0.618 + 1.90i)7-s + (0.309 + 0.951i)8-s + (2.30 − 1.67i)9-s + 3.23·10-s + (0.309 + 3.30i)11-s + 0.381·12-s + (−1 + 0.726i)13-s + (−0.618 − 1.90i)14-s + (0.381 − 1.17i)15-s + (−0.809 − 0.587i)16-s + (0.5 + 0.363i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.0681 + 0.209i)3-s + (0.154 − 0.475i)4-s + (−1.17 − 0.850i)5-s + (−0.126 − 0.0916i)6-s + (−0.233 + 0.718i)7-s + (0.109 + 0.336i)8-s + (0.769 − 0.559i)9-s + 1.02·10-s + (0.0931 + 0.995i)11-s + 0.110·12-s + (−0.277 + 0.201i)13-s + (−0.165 − 0.508i)14-s + (0.0986 − 0.303i)15-s + (−0.202 − 0.146i)16-s + (0.121 + 0.0881i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22\)    =    \(2 \cdot 11\)
Sign: $0.927 - 0.374i$
Analytic conductor: \(0.175670\)
Root analytic conductor: \(0.419131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{22} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 22,\ (\ :1/2),\ 0.927 - 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.459919 + 0.0893326i\)
\(L(\frac12)\) \(\approx\) \(0.459919 + 0.0893326i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (-0.309 - 3.30i)T \)
good3 \( 1 + (-0.118 - 0.363i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (2.61 + 1.90i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.618 - 1.90i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (1 - 0.726i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.5 - 0.363i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.80 + 5.56i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 1.23T + 23T^{2} \)
29 \( 1 + (-1.38 + 4.25i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.61 - 1.17i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.14 - 3.52i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.73 - 5.34i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 8.56T + 43T^{2} \)
47 \( 1 + (2 + 6.15i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.23 + 0.898i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.66 - 8.19i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2 - 1.45i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 11.0T + 67T^{2} \)
71 \( 1 + (-4.23 - 3.07i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.20 + 9.87i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (10.8 - 7.88i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (7.54 + 5.48i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 8.09T + 89T^{2} \)
97 \( 1 + (5.78 - 4.20i)T + (29.9 - 92.2i)T^{2} \)
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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

−18.17756986172014387027382435622, −16.82411981911449626733863598771, −15.54777422558285971247884950111, −15.16179757497127178864373049691, −12.78371803078076126247183040055, −11.73196195975361073076144697363, −9.732900602864643249518309140274, −8.578299937453888876233453362782, −7.02176996947940726846070608250, −4.60558888920413881633024178031, 3.65206342848791883968556909270, 7.05504734034573282193115387091, 8.119000129251193742156669325387, 10.26581297587051140598940030101, 11.19015273008704886636155042019, 12.65389133375843142327103425648, 14.21927936304491680625721920771, 15.78808119532245846683109328974, 16.77678219891911949871257985517, 18.53323601488478753406428324084

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