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Properties

Label 2-21e2-63.47-c1-0-15
Degree $2$
Conductor $441$
Sign $0.449 - 0.893i$
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  + (−0.105 + 0.0611i)2-s + (1.73 − 0.0838i)3-s + (−0.992 + 1.71i)4-s − 0.529·5-s + (−0.178 + 0.114i)6-s − 0.487i·8-s + (2.98 − 0.290i)9-s + (0.0560 − 0.0323i)10-s + 4.20i·11-s + (−1.57 + 3.05i)12-s + (1.74 − 1.00i)13-s + (−0.915 + 0.0444i)15-s + (−1.95 − 3.38i)16-s + (2.19 + 3.79i)17-s + (−0.298 + 0.213i)18-s + (4.54 + 2.62i)19-s + ⋯
L(s)  = 1  + (−0.0749 + 0.0432i)2-s + (0.998 − 0.0484i)3-s + (−0.496 + 0.859i)4-s − 0.236·5-s + (−0.0727 + 0.0468i)6-s − 0.172i·8-s + (0.995 − 0.0967i)9-s + (0.0177 − 0.0102i)10-s + 1.26i·11-s + (−0.454 + 0.882i)12-s + (0.484 − 0.279i)13-s + (−0.236 + 0.0114i)15-s + (−0.488 − 0.846i)16-s + (0.532 + 0.921i)17-s + (−0.0703 + 0.0503i)18-s + (1.04 + 0.601i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.449 - 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.449 - 0.893i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.449 - 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38905 + 0.855597i\)
\(L(\frac12)\) \(\approx\) \(1.38905 + 0.855597i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.73 + 0.0838i)T \)
7 \( 1 \)
good2 \( 1 + (0.105 - 0.0611i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + 0.529T + 5T^{2} \)
11 \( 1 - 4.20iT - 11T^{2} \)
13 \( 1 + (-1.74 + 1.00i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.19 - 3.79i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.54 - 2.62i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 6.27iT - 23T^{2} \)
29 \( 1 + (7.27 + 4.20i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.03 - 0.595i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.61 + 2.79i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.0994 - 0.172i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.96 + 6.86i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.98 + 8.63i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.65 - 2.10i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.71 + 11.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-11.3 + 6.55i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.29 + 5.69i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.50iT - 71T^{2} \)
73 \( 1 + (4.86 - 2.80i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.286 + 0.495i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.42 - 9.39i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.43 + 11.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.493 + 0.285i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.45205176002768435387137380856, −9.854632887305571340450540318274, −9.597682628478672240087153746979, −8.361062649043522858641711601959, −7.76204585800264796301598740996, −7.12860245404230854187802235079, −5.44144080209036532131115261049, −3.97439823923337406152638347297, −3.53413755294744091034200806444, −1.91773903380148788422866370649, 1.08090340569352673696338980292, 2.81276582880084920461016828813, 3.96457892952831906980325860328, 5.12068336499668059685279764614, 6.22421749007465921504915089718, 7.46334040651890504430822866606, 8.441028420385909415888126422705, 9.169491309902542080271484126316, 9.831770188334186500252896516810, 10.88591387659476334946695371019

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