Properties

Label 2-3e6-243.151-c1-0-4
Degree $2$
Conductor $729$
Sign $-0.310 - 0.950i$
Analytic cond. $5.82109$
Root an. cond. $2.41269$
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  + (−1.17 + 0.327i)2-s + (−0.436 + 0.263i)4-s + (−3.30 − 0.128i)5-s + (3.02 − 0.473i)7-s + (2.10 − 2.22i)8-s + (3.93 − 0.931i)10-s + (−0.345 + 2.53i)11-s + (−2.19 − 3.20i)13-s + (−3.40 + 1.54i)14-s + (−1.26 + 2.40i)16-s + (2.28 + 1.14i)17-s + (−0.0846 + 1.45i)19-s + (1.47 − 0.815i)20-s + (−0.421 − 3.08i)22-s + (−1.73 − 4.50i)23-s + ⋯
L(s)  = 1  + (−0.831 + 0.231i)2-s + (−0.218 + 0.131i)4-s + (−1.47 − 0.0574i)5-s + (1.14 − 0.179i)7-s + (0.743 − 0.787i)8-s + (1.24 − 0.294i)10-s + (−0.104 + 0.762i)11-s + (−0.609 − 0.889i)13-s + (−0.910 + 0.413i)14-s + (−0.316 + 0.601i)16-s + (0.554 + 0.278i)17-s + (−0.0194 + 0.333i)19-s + (0.330 − 0.182i)20-s + (−0.0899 − 0.658i)22-s + (−0.362 − 0.938i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.310 - 0.950i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.310 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.286204 + 0.394580i\)
\(L(\frac12)\) \(\approx\) \(0.286204 + 0.394580i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (1.17 - 0.327i)T + (1.71 - 1.03i)T^{2} \)
5 \( 1 + (3.30 + 0.128i)T + (4.98 + 0.387i)T^{2} \)
7 \( 1 + (-3.02 + 0.473i)T + (6.66 - 2.13i)T^{2} \)
11 \( 1 + (0.345 - 2.53i)T + (-10.5 - 2.94i)T^{2} \)
13 \( 1 + (2.19 + 3.20i)T + (-4.68 + 12.1i)T^{2} \)
17 \( 1 + (-2.28 - 1.14i)T + (10.1 + 13.6i)T^{2} \)
19 \( 1 + (0.0846 - 1.45i)T + (-18.8 - 2.20i)T^{2} \)
23 \( 1 + (1.73 + 4.50i)T + (-17.0 + 15.4i)T^{2} \)
29 \( 1 + (-4.11 + 2.93i)T + (9.38 - 27.4i)T^{2} \)
31 \( 1 + (-5.18 - 5.94i)T + (-4.19 + 30.7i)T^{2} \)
37 \( 1 + (5.96 - 8.00i)T + (-10.6 - 35.4i)T^{2} \)
41 \( 1 + (-1.91 - 7.44i)T + (-35.8 + 19.8i)T^{2} \)
43 \( 1 + (8.37 - 7.59i)T + (4.16 - 42.7i)T^{2} \)
47 \( 1 + (4.91 - 5.63i)T + (-6.36 - 46.5i)T^{2} \)
53 \( 1 + (-0.683 - 3.87i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (1.90 + 0.779i)T + (42.1 + 41.3i)T^{2} \)
61 \( 1 + (-7.42 - 4.48i)T + (28.4 + 53.9i)T^{2} \)
67 \( 1 + (6.69 + 4.78i)T + (21.6 + 63.3i)T^{2} \)
71 \( 1 + (2.48 - 8.30i)T + (-59.3 - 39.0i)T^{2} \)
73 \( 1 + (2.45 + 0.580i)T + (65.2 + 32.7i)T^{2} \)
79 \( 1 + (-7.41 - 7.27i)T + (1.53 + 78.9i)T^{2} \)
83 \( 1 + (-2.15 + 8.37i)T + (-72.6 - 40.0i)T^{2} \)
89 \( 1 + (-4.79 - 16.0i)T + (-74.3 + 48.9i)T^{2} \)
97 \( 1 + (6.93 - 0.269i)T + (96.7 - 7.51i)T^{2} \)
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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

−10.36946986248341392378204183661, −9.936114725804616092652530594512, −8.459807329517686250543060934862, −8.099773558144518434306771877352, −7.66438099755273665041415872609, −6.66791889759925137391174803357, −4.78965957900881878445286014652, −4.50326707855559487089710024522, −3.17418248960914301456749072212, −1.18420142394264700803961027818, 0.40609026652720932437522199889, 1.94977569553314281478881398343, 3.61930044749731403332477605842, 4.64158021313711344656353644235, 5.39502625817300408044635991795, 7.08627896875105066944848056429, 7.81796582389890312852265453994, 8.420873158718039609813481954378, 9.086281280574294614412825828366, 10.18521316338281738104707735858

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