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Properties

Label 2-21e2-63.47-c1-0-23
Degree $2$
Conductor $441$
Sign $0.987 + 0.156i$
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  + (−1.02 + 0.592i)2-s + (1.25 − 1.19i)3-s + (−0.296 + 0.514i)4-s + 2.83·5-s + (−0.576 + 1.97i)6-s − 3.07i·8-s + (0.136 − 2.99i)9-s + (−2.91 + 1.68i)10-s − 0.157i·11-s + (0.243 + 0.998i)12-s + (3.41 − 1.97i)13-s + (3.55 − 3.39i)15-s + (1.23 + 2.13i)16-s + (2.07 + 3.58i)17-s + (1.63 + 3.15i)18-s + (−5.48 − 3.16i)19-s + ⋯
L(s)  = 1  + (−0.726 + 0.419i)2-s + (0.723 − 0.690i)3-s + (−0.148 + 0.257i)4-s + 1.26·5-s + (−0.235 + 0.804i)6-s − 1.08i·8-s + (0.0455 − 0.998i)9-s + (−0.921 + 0.532i)10-s − 0.0475i·11-s + (0.0702 + 0.288i)12-s + (0.947 − 0.546i)13-s + (0.917 − 0.876i)15-s + (0.307 + 0.532i)16-s + (0.502 + 0.870i)17-s + (0.385 + 0.744i)18-s + (−1.25 − 0.726i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.156i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.987 + 0.156i$
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: \((2,\ 441,\ (\ :1/2),\ 0.987 + 0.156i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41176 - 0.110861i\)
\(L(\frac12)\) \(\approx\) \(1.41176 - 0.110861i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.25 + 1.19i)T \)
7 \( 1 \)
good2 \( 1 + (1.02 - 0.592i)T + (1 - 1.73i)T^{2} \)
5 \( 1 - 2.83T + 5T^{2} \)
11 \( 1 + 0.157iT - 11T^{2} \)
13 \( 1 + (-3.41 + 1.97i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.07 - 3.58i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.48 + 3.16i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.546iT - 23T^{2} \)
29 \( 1 + (-4.02 - 2.32i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.112 - 0.0647i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.23 + 2.13i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.99 - 3.45i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.28 + 5.68i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.33 - 7.50i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.25 + 1.30i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.80 - 3.12i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.91 - 1.68i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.663 - 1.14i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.409iT - 71T^{2} \)
73 \( 1 + (13.0 - 7.50i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.16 + 3.74i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.22 - 5.58i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.52 + 4.37i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.18 - 1.26i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.74339214920758358367899048130, −9.936127219393097931020734400232, −8.964351633568676892917663762215, −8.541493138272205988671495783822, −7.58960516391295824517915179497, −6.53242522797405565957472986242, −5.89188827503633216180181398543, −4.02664186352272078704857493431, −2.72392285039531164903705933360, −1.27176810243127001502962439950, 1.63261154579578451543353117917, 2.63962178663827384446069399174, 4.24939471607639156704128919698, 5.39961399265391999010384902423, 6.31283406646851430402841387228, 7.938366607768720653242127957409, 8.848020311395260137159427891981, 9.353921929634267864162157689427, 10.20313503477002926434578854587, 10.60000190660181443561135114094

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