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Properties

Label 2-21e2-21.20-c3-0-38
Degree $2$
Conductor $441$
Sign $-0.0980 - 0.995i$
Analytic cond. $26.0198$
Root an. cond. $5.10096$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.52i·2-s − 12.4·4-s − 1.26·5-s + 20.1i·8-s + 5.72i·10-s − 41.5i·11-s − 85.7i·13-s − 8.54·16-s + 77.7·17-s − 48.6i·19-s + 15.7·20-s − 188.·22-s + 90.9i·23-s − 123.·25-s − 387.·26-s + ⋯
L(s)  = 1  − 1.59i·2-s − 1.55·4-s − 0.113·5-s + 0.890i·8-s + 0.181i·10-s − 1.14i·11-s − 1.82i·13-s − 0.133·16-s + 1.10·17-s − 0.587i·19-s + 0.176·20-s − 1.82·22-s + 0.824i·23-s − 0.987·25-s − 2.92·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0980 - 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0980 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.0980 - 0.995i$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (440, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ -0.0980 - 0.995i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9045550385\)
\(L(\frac12)\) \(\approx\) \(0.9045550385\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good2 \( 1 + 4.52iT - 8T^{2} \)
5 \( 1 + 1.26T + 125T^{2} \)
11 \( 1 + 41.5iT - 1.33e3T^{2} \)
13 \( 1 + 85.7iT - 2.19e3T^{2} \)
17 \( 1 - 77.7T + 4.91e3T^{2} \)
19 \( 1 + 48.6iT - 6.85e3T^{2} \)
23 \( 1 - 90.9iT - 1.21e4T^{2} \)
29 \( 1 - 151. iT - 2.43e4T^{2} \)
31 \( 1 - 88.1iT - 2.97e4T^{2} \)
37 \( 1 - 90.5T + 5.06e4T^{2} \)
41 \( 1 + 383.T + 6.89e4T^{2} \)
43 \( 1 + 227.T + 7.95e4T^{2} \)
47 \( 1 - 139.T + 1.03e5T^{2} \)
53 \( 1 - 334. iT - 1.48e5T^{2} \)
59 \( 1 + 880.T + 2.05e5T^{2} \)
61 \( 1 + 13.1iT - 2.26e5T^{2} \)
67 \( 1 + 442.T + 3.00e5T^{2} \)
71 \( 1 + 341. iT - 3.57e5T^{2} \)
73 \( 1 + 921. iT - 3.89e5T^{2} \)
79 \( 1 + 413.T + 4.93e5T^{2} \)
83 \( 1 - 954.T + 5.71e5T^{2} \)
89 \( 1 + 29.6T + 7.04e5T^{2} \)
97 \( 1 + 1.19e3iT - 9.12e5T^{2} \)
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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

−10.37929263594421387156230443591, −9.445854536798516418237781032274, −8.467268105657158133833371192251, −7.55997093984052788201735433319, −5.92046215549100461721717567206, −4.98096641765911498856288108261, −3.39069189117067065760146952800, −3.10640221023999586996285041371, −1.40593536985928911933323742898, −0.30318481579374790167520254007, 1.94708686818213920740037993911, 4.02306902609063975798512460884, 4.80239384151850640446575384786, 5.97468807403038224240909089622, 6.76534852448736310224064727350, 7.55552488628339324747328701254, 8.335862256783987464805779173364, 9.403175912541014724560637839311, 10.02247709910330524375745340026, 11.57338570662302151753333662758

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