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Properties

Label 2-21e2-63.59-c1-0-19
Degree $2$
Conductor $441$
Sign $0.251 - 0.967i$
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  + (1.97 + 1.13i)2-s + (1.70 + 0.315i)3-s + (1.59 + 2.75i)4-s − 1.43·5-s + (2.99 + 2.55i)6-s + 2.69i·8-s + (2.80 + 1.07i)9-s + (−2.82 − 1.63i)10-s + 3.23i·11-s + (1.84 + 5.19i)12-s + (−4.43 − 2.55i)13-s + (−2.44 − 0.451i)15-s + (0.119 − 0.207i)16-s + (0.545 − 0.945i)17-s + (4.30 + 5.30i)18-s + (3.88 − 2.24i)19-s + ⋯
L(s)  = 1  + (1.39 + 0.804i)2-s + (0.983 + 0.181i)3-s + (0.795 + 1.37i)4-s − 0.641·5-s + (1.22 + 1.04i)6-s + 0.951i·8-s + (0.933 + 0.357i)9-s + (−0.894 − 0.516i)10-s + 0.975i·11-s + (0.531 + 1.49i)12-s + (−1.22 − 0.709i)13-s + (−0.630 − 0.116i)15-s + (0.0298 − 0.0517i)16-s + (0.132 − 0.229i)17-s + (1.01 + 1.25i)18-s + (0.891 − 0.514i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.251 - 0.967i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.251 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.73664 + 2.11548i\)
\(L(\frac12)\) \(\approx\) \(2.73664 + 2.11548i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.70 - 0.315i)T \)
7 \( 1 \)
good2 \( 1 + (-1.97 - 1.13i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 1.43T + 5T^{2} \)
11 \( 1 - 3.23iT - 11T^{2} \)
13 \( 1 + (4.43 + 2.55i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.545 + 0.945i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.88 + 2.24i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.00iT - 23T^{2} \)
29 \( 1 + (-1.02 + 0.593i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.24 - 1.87i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.119 - 0.207i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.71 - 6.43i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.82 + 6.62i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.11 + 3.65i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.07 + 3.50i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.73 - 8.20i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.82 + 1.63i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.330 + 0.571i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.82iT - 71T^{2} \)
73 \( 1 + (6.33 + 3.65i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.83 - 3.16i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.45 - 9.44i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.84 - 11.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.69 - 1.55i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.83292642391312070541009461347, −10.27207731553294927796414215533, −9.488839239499246107235483286220, −8.136186589603559734464636485147, −7.42808961844174815463814004329, −6.85913005904118611646841251544, −5.22969448686083740174150497210, −4.59099695571595658706851127418, −3.56820859288581742935215164302, −2.55841411561758779768680458570, 1.78151824623716981036768266388, 3.07451979381195402757334272721, 3.74419555673563049856803413216, 4.75189961905071882743011751470, 5.93051441611931614744753654415, 7.26683170611389871172919088399, 8.082724807925971322665475943917, 9.278673523036685913146387489316, 10.18430360443818220009081062979, 11.37141891685722301508730533041

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