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Properties

Label 2-3e6-27.25-c1-0-16
Degree $2$
Conductor $729$
Sign $0.993 - 0.116i$
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  + (−0.426 + 2.42i)2-s + (−3.79 − 1.38i)4-s + (−2.35 + 1.97i)5-s + (2.49 − 0.909i)7-s + (2.50 − 4.34i)8-s + (−3.78 − 6.55i)10-s + (−2.63 − 2.20i)11-s + (−0.580 − 3.29i)13-s + (1.13 + 6.43i)14-s + (3.25 + 2.72i)16-s + (−1.28 − 2.22i)17-s + (1.04 − 1.81i)19-s + (11.6 − 4.25i)20-s + (6.46 − 5.42i)22-s + (−0.502 − 0.182i)23-s + ⋯
L(s)  = 1  + (−0.301 + 1.71i)2-s + (−1.89 − 0.691i)4-s + (−1.05 + 0.885i)5-s + (0.944 − 0.343i)7-s + (0.886 − 1.53i)8-s + (−1.19 − 2.07i)10-s + (−0.793 − 0.665i)11-s + (−0.161 − 0.913i)13-s + (0.303 + 1.71i)14-s + (0.813 + 0.682i)16-s + (−0.311 − 0.540i)17-s + (0.240 − 0.416i)19-s + (2.61 − 0.951i)20-s + (1.37 − 1.15i)22-s + (−0.104 − 0.0381i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.993 - 0.116i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.993 - 0.116i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.488638 + 0.0284599i\)
\(L(\frac12)\) \(\approx\) \(0.488638 + 0.0284599i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.426 - 2.42i)T + (-1.87 - 0.684i)T^{2} \)
5 \( 1 + (2.35 - 1.97i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (-2.49 + 0.909i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (2.63 + 2.20i)T + (1.91 + 10.8i)T^{2} \)
13 \( 1 + (0.580 + 3.29i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (1.28 + 2.22i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.04 + 1.81i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.502 + 0.182i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.439 + 2.49i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (7.24 + 2.63i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (-5.14 - 8.91i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.848 + 4.81i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-2.10 - 1.76i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-5.31 + 1.93i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 - 6.42T + 53T^{2} \)
59 \( 1 + (-1.26 + 1.06i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (13.5 - 4.91i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.02 + 5.78i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (7.40 + 12.8i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.940 - 1.62i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.98 + 16.9i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-0.689 + 3.90i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (2.54 - 4.41i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.14 + 6.83i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.42162844681983859357660440416, −9.168683068546513891248598512946, −8.204276216775787799708805390260, −7.64828319490062083662020211608, −7.28317172309611311079116292360, −6.12546698689150273333907353153, −5.21103084414943631865309084208, −4.32397111431135198842834675649, −2.98110128860672980408828884439, −0.30044551073686893832044183833, 1.39821524699995410657806521834, 2.38866165982343432024543814065, 3.90126081637754393294400078378, 4.46204165922005192941877541063, 5.36472288442885385514698113824, 7.34060587679714591712096478936, 8.181595817088048681002437708343, 8.835768682873583936755351759222, 9.563181708580393455938539686548, 10.67070560273723737466581606802

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