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Properties

Label 2-21e2-9.4-c1-0-1
Degree $2$
Conductor $441$
Sign $0.5 - 0.866i$
Analytic cond. $3.52140$
Root an. cond. $1.87654$
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  + (−1.26 − 2.19i)2-s + (1.11 + 1.32i)3-s + (−2.20 + 3.82i)4-s + (−0.439 + 0.761i)5-s + (1.49 − 4.12i)6-s + 6.10·8-s + (−0.520 + 2.95i)9-s + 2.22·10-s + (−1.93 − 3.35i)11-s + (−7.52 + 1.32i)12-s + (−2.72 + 4.72i)13-s + (−1.5 + 0.264i)15-s + (−3.31 − 5.74i)16-s − 1.65·17-s + (7.13 − 2.59i)18-s − 2.41·19-s + ⋯
L(s)  = 1  + (−0.895 − 1.55i)2-s + (0.642 + 0.766i)3-s + (−1.10 + 1.91i)4-s + (−0.196 + 0.340i)5-s + (0.612 − 1.68i)6-s + 2.15·8-s + (−0.173 + 0.984i)9-s + 0.704·10-s + (−0.584 − 1.01i)11-s + (−2.17 + 0.383i)12-s + (−0.756 + 1.30i)13-s + (−0.387 + 0.0682i)15-s + (−0.829 − 1.43i)16-s − 0.400·17-s + (1.68 − 0.612i)18-s − 0.553·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.5 - 0.866i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.5 - 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.500547 + 0.288991i\)
\(L(\frac12)\) \(\approx\) \(0.500547 + 0.288991i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.11 - 1.32i)T \)
7 \( 1 \)
good2 \( 1 + (1.26 + 2.19i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.439 - 0.761i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.93 + 3.35i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.72 - 4.72i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.65T + 17T^{2} \)
19 \( 1 + 2.41T + 19T^{2} \)
23 \( 1 + (1.58 - 2.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.02 - 5.23i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.27 - 3.94i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.55T + 37T^{2} \)
41 \( 1 + (0.592 - 1.02i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.0923 + 0.160i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.511 + 0.885i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 7.29T + 53T^{2} \)
59 \( 1 + (-3.33 + 5.76i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.29 + 2.24i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.47 + 2.56i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.68T + 71T^{2} \)
73 \( 1 - 12.7T + 73T^{2} \)
79 \( 1 + (-2.97 - 5.15i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.109 + 0.189i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 11.0T + 89T^{2} \)
97 \( 1 + (-6.25 - 10.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.98851669110139425411338446369, −10.48839129333782626797017026280, −9.540656354503901823172328840728, −8.869043690466518415809044086479, −8.206664961532715655922637115499, −7.03946518869977314404889371063, −5.04352777388769430097390418726, −3.85946546189282595522289114244, −3.03559974975142234966392242999, −1.96793657960726988392560763943, 0.43487204845687675740811970151, 2.38824226298491302348913423670, 4.49201352824221679105927262574, 5.62314904051955695844504743676, 6.66560489808404267133224791144, 7.45952123708261193623007761004, 8.097039250263532918910944769656, 8.714407243730658374011374379466, 9.777529359126568182616837949062, 10.40554397197355792849498176923

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