Properties

Label 2-21e2-147.131-c1-0-7
Degree $2$
Conductor $441$
Sign $0.589 - 0.807i$
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  + (−2.45 + 0.962i)2-s + (3.61 − 3.35i)4-s + (1.47 − 1.00i)5-s + (2.09 + 1.60i)7-s + (−3.35 + 6.97i)8-s + (−2.65 + 3.89i)10-s + (−0.538 + 3.57i)11-s + (−2.76 − 2.20i)13-s + (−6.69 − 1.92i)14-s + (0.785 − 10.4i)16-s + (2.32 − 0.718i)17-s + (6.52 + 3.76i)19-s + (1.96 − 8.60i)20-s + (−2.11 − 9.28i)22-s + (1.91 − 6.21i)23-s + ⋯
L(s)  = 1  + (−1.73 + 0.680i)2-s + (1.80 − 1.67i)4-s + (0.660 − 0.450i)5-s + (0.793 + 0.608i)7-s + (−1.18 + 2.46i)8-s + (−0.839 + 1.23i)10-s + (−0.162 + 1.07i)11-s + (−0.767 − 0.612i)13-s + (−1.79 − 0.514i)14-s + (0.196 − 2.61i)16-s + (0.564 − 0.174i)17-s + (1.49 + 0.864i)19-s + (0.439 − 1.92i)20-s + (−0.451 − 1.97i)22-s + (0.399 − 1.29i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.589 - 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.589 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.589 - 0.807i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (278, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.589 - 0.807i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.681872 + 0.346579i\)
\(L(\frac12)\) \(\approx\) \(0.681872 + 0.346579i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-2.09 - 1.60i)T \)
good2 \( 1 + (2.45 - 0.962i)T + (1.46 - 1.36i)T^{2} \)
5 \( 1 + (-1.47 + 1.00i)T + (1.82 - 4.65i)T^{2} \)
11 \( 1 + (0.538 - 3.57i)T + (-10.5 - 3.24i)T^{2} \)
13 \( 1 + (2.76 + 2.20i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (-2.32 + 0.718i)T + (14.0 - 9.57i)T^{2} \)
19 \( 1 + (-6.52 - 3.76i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.91 + 6.21i)T + (-19.0 - 12.9i)T^{2} \)
29 \( 1 + (-4.35 - 0.994i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + (5.85 - 3.38i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.18 + 3.88i)T + (2.76 + 36.8i)T^{2} \)
41 \( 1 + (-3.53 - 1.70i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-3.05 + 1.47i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-1.81 - 4.62i)T + (-34.4 + 31.9i)T^{2} \)
53 \( 1 + (3.20 + 3.45i)T + (-3.96 + 52.8i)T^{2} \)
59 \( 1 + (-9.48 - 6.46i)T + (21.5 + 54.9i)T^{2} \)
61 \( 1 + (-4.54 + 4.89i)T + (-4.55 - 60.8i)T^{2} \)
67 \( 1 + (-6.90 - 11.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.21 + 0.276i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (-8.19 - 3.21i)T + (53.5 + 49.6i)T^{2} \)
79 \( 1 + (-5.24 + 9.07i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.72 + 10.9i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (7.03 - 1.05i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 - 1.36iT - 97T^{2} \)
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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

−10.80578973766684947216764795355, −9.976687619017131872529208238758, −9.442862195520560586826669934679, −8.563374063679109061875853956146, −7.71016908221751772073097375684, −7.04762819225558357258085698646, −5.63357653197596803156084320272, −5.12726545901711607739738050275, −2.42631048724440734956397722575, −1.29386798142018723481165912296, 1.01955806357107036947473238175, 2.32538855539676354314170316185, 3.48203754079619802958359475562, 5.38373802533072879687071359512, 6.85465916065293812124175187080, 7.55071994696778969735998620274, 8.381236741734963746115186063039, 9.468710693341684246498250026868, 9.907089464499505032670923721497, 10.98078525139123061051936459664

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