Properties

Label 2-3e6-81.31-c1-0-13
Degree $2$
Conductor $729$
Sign $0.494 - 0.868i$
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  + (0.280 + 0.649i)2-s + (1.02 − 1.09i)4-s + (−0.199 + 3.42i)5-s + (2.63 − 0.623i)7-s + (2.32 + 0.846i)8-s + (−2.28 + 0.831i)10-s + (−1.23 + 0.811i)11-s + (5.51 − 0.644i)13-s + (1.14 + 1.53i)14-s + (−0.0725 − 1.24i)16-s + (−3.86 − 3.24i)17-s + (−3.94 + 3.30i)19-s + (3.53 + 3.74i)20-s + (−0.873 − 0.574i)22-s + (1.68 + 0.399i)23-s + ⋯
L(s)  = 1  + (0.198 + 0.459i)2-s + (0.514 − 0.545i)4-s + (−0.0893 + 1.53i)5-s + (0.994 − 0.235i)7-s + (0.822 + 0.299i)8-s + (−0.722 + 0.262i)10-s + (−0.372 + 0.244i)11-s + (1.52 − 0.178i)13-s + (0.305 + 0.410i)14-s + (−0.0181 − 0.311i)16-s + (−0.937 − 0.786i)17-s + (−0.904 + 0.758i)19-s + (0.790 + 0.837i)20-s + (−0.186 − 0.122i)22-s + (0.351 + 0.0833i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.494 - 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.494 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.494 - 0.868i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (676, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.494 - 0.868i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88892 + 1.09809i\)
\(L(\frac12)\) \(\approx\) \(1.88892 + 1.09809i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.280 - 0.649i)T + (-1.37 + 1.45i)T^{2} \)
5 \( 1 + (0.199 - 3.42i)T + (-4.96 - 0.580i)T^{2} \)
7 \( 1 + (-2.63 + 0.623i)T + (6.25 - 3.14i)T^{2} \)
11 \( 1 + (1.23 - 0.811i)T + (4.35 - 10.1i)T^{2} \)
13 \( 1 + (-5.51 + 0.644i)T + (12.6 - 2.99i)T^{2} \)
17 \( 1 + (3.86 + 3.24i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (3.94 - 3.30i)T + (3.29 - 18.7i)T^{2} \)
23 \( 1 + (-1.68 - 0.399i)T + (20.5 + 10.3i)T^{2} \)
29 \( 1 + (-1.28 + 1.72i)T + (-8.31 - 27.7i)T^{2} \)
31 \( 1 + (-1.09 - 3.64i)T + (-25.9 + 17.0i)T^{2} \)
37 \( 1 + (0.891 - 5.05i)T + (-34.7 - 12.6i)T^{2} \)
41 \( 1 + (1.78 - 4.14i)T + (-28.1 - 29.8i)T^{2} \)
43 \( 1 + (-0.736 - 0.370i)T + (25.6 + 34.4i)T^{2} \)
47 \( 1 + (-1.92 + 6.43i)T + (-39.2 - 25.8i)T^{2} \)
53 \( 1 + (-2.58 + 4.48i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.95 - 1.94i)T + (23.3 + 54.1i)T^{2} \)
61 \( 1 + (-3.71 - 3.94i)T + (-3.54 + 60.8i)T^{2} \)
67 \( 1 + (3.91 + 5.25i)T + (-19.2 + 64.1i)T^{2} \)
71 \( 1 + (-3.30 + 1.20i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-0.668 - 0.243i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (6.21 + 14.4i)T + (-54.2 + 57.4i)T^{2} \)
83 \( 1 + (-2.35 - 5.45i)T + (-56.9 + 60.3i)T^{2} \)
89 \( 1 + (3.78 + 1.37i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-0.700 - 12.0i)T + (-96.3 + 11.2i)T^{2} \)
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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

−10.70245496951857951815948583781, −10.06255054911948469649607743901, −8.540675887450069334096357953306, −7.76372589539509417865559032815, −6.82099083518734536630062833103, −6.38117183918477818027597256402, −5.29330371909516154034940934764, −4.17420911500723743365889250285, −2.83355231997359164681245153646, −1.67600786416095111827899047249, 1.25111789012192581166268510287, 2.28458947642796521904522559839, 3.93275696336419853981170890809, 4.52476572030650732210229395341, 5.60408107636722846330581024230, 6.71750633432003784388768777195, 8.040253785984663832745601488566, 8.495267812100355547323028019189, 9.064726103870312949391946248766, 10.71328363857410887383530447051

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