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Properties

Label 2-21e2-9.4-c1-0-19
Degree $2$
Conductor $441$
Sign $0.927 - 0.373i$
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  + (0.119 + 0.207i)2-s + (0.619 + 1.61i)3-s + (0.971 − 1.68i)4-s + (0.590 − 1.02i)5-s + (−0.260 + 0.321i)6-s + 0.942·8-s + (−2.23 + 2.00i)9-s + 0.282·10-s + (1.85 + 3.20i)11-s + (3.32 + 0.528i)12-s + (0.5 − 0.866i)13-s + (2.02 + 0.321i)15-s + (−1.83 − 3.16i)16-s + 6.94·17-s + (−0.681 − 0.222i)18-s − 1.94·19-s + ⋯
L(s)  = 1  + (0.0845 + 0.146i)2-s + (0.357 + 0.933i)3-s + (0.485 − 0.841i)4-s + (0.264 − 0.457i)5-s + (−0.106 + 0.131i)6-s + 0.333·8-s + (−0.744 + 0.668i)9-s + 0.0893·10-s + (0.558 + 0.967i)11-s + (0.959 + 0.152i)12-s + (0.138 − 0.240i)13-s + (0.522 + 0.0830i)15-s + (−0.457 − 0.792i)16-s + 1.68·17-s + (−0.160 − 0.0524i)18-s − 0.445·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.927 - 0.373i$
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: \((2,\ 441,\ (\ :1/2),\ 0.927 - 0.373i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90868 + 0.369602i\)
\(L(\frac12)\) \(\approx\) \(1.90868 + 0.369602i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.619 - 1.61i)T \)
7 \( 1 \)
good2 \( 1 + (-0.119 - 0.207i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.590 + 1.02i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.85 - 3.20i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 6.94T + 17T^{2} \)
19 \( 1 + 1.94T + 19T^{2} \)
23 \( 1 + (-2.80 + 4.85i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.119 + 0.207i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.830 + 1.43i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 9.54T + 37T^{2} \)
41 \( 1 + (5.09 - 8.81i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.11 + 1.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.91 - 5.04i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 11.6T + 53T^{2} \)
59 \( 1 + (-1.30 + 2.25i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.80 + 6.58i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.75 - 3.03i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.60T + 71T^{2} \)
73 \( 1 + 15.1T + 73T^{2} \)
79 \( 1 + (3.68 + 6.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.47 + 6.01i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 2.74T + 89T^{2} \)
97 \( 1 + (-3.58 - 6.20i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.92365505224322144853468831435, −10.12994862207234494642669672636, −9.599565862782728349861443974019, −8.658317131670283147630531209756, −7.50066441723692545251114393848, −6.32945670162871321367334806424, −5.27475152446301161997010811146, −4.60010295392169045779689100256, −3.13445257034930462443494352841, −1.59441615894190796634440062062, 1.56343346139272356099306735153, 2.98066055790184014548705809692, 3.62784330832784873183962070791, 5.62760298242770127860157479205, 6.62509372449290444744929532947, 7.30954268820007377741395611614, 8.268057849647678802039521716201, 8.962465969896783485277625413886, 10.33517431281013903834677105462, 11.31721323793576656884781729641

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