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Properties

Label 2-3e6-27.22-c1-0-27
Degree $2$
Conductor $729$
Sign $-0.835 + 0.549i$
Analytic cond. $5.82109$
Root an. cond. $2.41269$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.93 + 1.62i)2-s + (0.766 − 4.34i)4-s + (0.439 − 0.160i)5-s + (−0.560 − 3.17i)7-s + (3.05 + 5.28i)8-s + (−0.592 + 1.02i)10-s + (−2.91 − 1.06i)11-s + (−1.67 − 1.40i)13-s + (6.25 + 5.25i)14-s + (−6.23 − 2.27i)16-s + (1.5 − 2.59i)17-s + (−0.0209 − 0.0362i)19-s + (−0.358 − 2.03i)20-s + (7.39 − 2.68i)22-s + (−1.06 + 6.01i)23-s + ⋯
L(s)  = 1  + (−1.37 + 1.15i)2-s + (0.383 − 2.17i)4-s + (0.196 − 0.0715i)5-s + (−0.211 − 1.20i)7-s + (1.07 + 1.86i)8-s + (−0.187 + 0.324i)10-s + (−0.880 − 0.320i)11-s + (−0.464 − 0.389i)13-s + (1.67 + 1.40i)14-s + (−1.55 − 0.567i)16-s + (0.363 − 0.630i)17-s + (−0.00480 − 0.00832i)19-s + (−0.0801 − 0.454i)20-s + (1.57 − 0.573i)22-s + (−0.221 + 1.25i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.835 + 0.549i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.835 + 0.549i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.93 - 1.62i)T + (0.347 - 1.96i)T^{2} \)
5 \( 1 + (-0.439 + 0.160i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (0.560 + 3.17i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (2.91 + 1.06i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (1.67 + 1.40i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0209 + 0.0362i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.06 - 6.01i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (5.03 - 4.22i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (1.08 - 6.13i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (1.79 - 3.11i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.90 + 4.95i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-0.553 - 0.201i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-1.67 - 9.51i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + 4.95T + 53T^{2} \)
59 \( 1 + (8.01 - 2.91i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (0.220 + 1.24i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-7.66 - 6.43i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-5.91 + 10.2i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.11 - 7.13i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.46 + 7.10i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-1.15 + 0.970i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (7.93 + 13.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (17.5 + 6.37i)T + (74.3 + 62.3i)T^{2} \)
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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

−9.831819747443668462178089259249, −9.223002987790039377060540649565, −8.103364587410407254375067348465, −7.46302507075974686141784968879, −6.97939869507714586158842520753, −5.74897436891187745832791951118, −5.07487139791385402105062047126, −3.36772107857940486959429754203, −1.44668256720594348858531417484, 0, 2.07119815925431077216697582968, 2.51527548959316819569282873996, 3.89053468481257972549392596174, 5.38815390458204314634861930472, 6.53375410811606645582484143018, 7.85165451612727577627793958826, 8.291777669695185238841733282024, 9.333943281223387126307697348570, 9.793515397392181082478464145257

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