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Properties

Label 2-3e6-81.34-c1-0-8
Degree $2$
Conductor $729$
Sign $-0.998 - 0.0613i$
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  + (−0.880 + 2.04i)2-s + (−2.01 − 2.13i)4-s + (0.171 + 2.94i)5-s + (2.87 + 0.681i)7-s + (1.96 − 0.714i)8-s + (−6.15 − 2.23i)10-s + (4.43 + 2.91i)11-s + (0.730 + 0.0853i)13-s + (−3.92 + 5.26i)14-s + (0.0720 − 1.23i)16-s + (1.11 − 0.933i)17-s + (4.21 + 3.53i)19-s + (5.94 − 6.29i)20-s + (−9.85 + 6.48i)22-s + (−0.438 + 0.103i)23-s + ⋯
L(s)  = 1  + (−0.622 + 1.44i)2-s + (−1.00 − 1.06i)4-s + (0.0765 + 1.31i)5-s + (1.08 + 0.257i)7-s + (0.693 − 0.252i)8-s + (−1.94 − 0.708i)10-s + (1.33 + 0.879i)11-s + (0.202 + 0.0236i)13-s + (−1.04 + 1.40i)14-s + (0.0180 − 0.309i)16-s + (0.269 − 0.226i)17-s + (0.966 + 0.810i)19-s + (1.32 − 1.40i)20-s + (−2.10 + 1.38i)22-s + (−0.0914 + 0.0216i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0613i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.998 - 0.0613i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.998 - 0.0613i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0394365 + 1.28504i\)
\(L(\frac12)\) \(\approx\) \(0.0394365 + 1.28504i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.880 - 2.04i)T + (-1.37 - 1.45i)T^{2} \)
5 \( 1 + (-0.171 - 2.94i)T + (-4.96 + 0.580i)T^{2} \)
7 \( 1 + (-2.87 - 0.681i)T + (6.25 + 3.14i)T^{2} \)
11 \( 1 + (-4.43 - 2.91i)T + (4.35 + 10.1i)T^{2} \)
13 \( 1 + (-0.730 - 0.0853i)T + (12.6 + 2.99i)T^{2} \)
17 \( 1 + (-1.11 + 0.933i)T + (2.95 - 16.7i)T^{2} \)
19 \( 1 + (-4.21 - 3.53i)T + (3.29 + 18.7i)T^{2} \)
23 \( 1 + (0.438 - 0.103i)T + (20.5 - 10.3i)T^{2} \)
29 \( 1 + (-2.12 - 2.85i)T + (-8.31 + 27.7i)T^{2} \)
31 \( 1 + (-0.319 + 1.06i)T + (-25.9 - 17.0i)T^{2} \)
37 \( 1 + (0.643 + 3.64i)T + (-34.7 + 12.6i)T^{2} \)
41 \( 1 + (4.05 + 9.38i)T + (-28.1 + 29.8i)T^{2} \)
43 \( 1 + (7.08 - 3.55i)T + (25.6 - 34.4i)T^{2} \)
47 \( 1 + (1.58 + 5.29i)T + (-39.2 + 25.8i)T^{2} \)
53 \( 1 + (1.83 + 3.17i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.98 - 3.93i)T + (23.3 - 54.1i)T^{2} \)
61 \( 1 + (2.98 - 3.16i)T + (-3.54 - 60.8i)T^{2} \)
67 \( 1 + (-6.75 + 9.07i)T + (-19.2 - 64.1i)T^{2} \)
71 \( 1 + (12.6 + 4.60i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-3.72 + 1.35i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-1.12 + 2.61i)T + (-54.2 - 57.4i)T^{2} \)
83 \( 1 + (-0.139 + 0.323i)T + (-56.9 - 60.3i)T^{2} \)
89 \( 1 + (-5.34 + 1.94i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-0.409 + 7.03i)T + (-96.3 - 11.2i)T^{2} \)
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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

−10.54213517551659822238488161989, −9.740416792835433107294419601676, −8.932259149505946580040662285718, −8.001501781756751856025735031414, −7.25116114875964801518140625936, −6.70356636858261267858334408563, −5.80191874198925807991981215181, −4.80875562640251683153786322180, −3.40781573630554051711551298566, −1.71997295530448616917943575954, 0.997777402832915122052052988521, 1.50166744937843276642666865480, 3.16585885389045233439465858786, 4.24937650798810599315203017920, 5.07806968042640228260925264678, 6.38578377828716027087756220032, 7.973541472845460694754491677604, 8.559978823355172931886224061253, 9.190364082358578569068132505816, 9.920550216268107466355433411393

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