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Properties

Label 2-3e6-27.13-c1-0-26
Degree $2$
Conductor $729$
Sign $-0.597 + 0.802i$
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  + (−0.135 − 0.765i)2-s + (1.31 − 0.477i)4-s + (−1.82 − 1.52i)5-s + (2.35 + 0.855i)7-s + (−1.32 − 2.28i)8-s + (−0.924 + 1.60i)10-s + (−2.40 + 2.01i)11-s + (0.232 − 1.31i)13-s + (0.337 − 1.91i)14-s + (0.564 − 0.473i)16-s + (3.13 − 5.43i)17-s + (−4.03 − 6.98i)19-s + (−3.11 − 1.13i)20-s + (1.87 + 1.57i)22-s + (3.81 − 1.38i)23-s + ⋯
L(s)  = 1  + (−0.0954 − 0.541i)2-s + (0.655 − 0.238i)4-s + (−0.814 − 0.683i)5-s + (0.888 + 0.323i)7-s + (−0.466 − 0.808i)8-s + (−0.292 + 0.506i)10-s + (−0.725 + 0.608i)11-s + (0.0643 − 0.365i)13-s + (0.0902 − 0.512i)14-s + (0.141 − 0.118i)16-s + (0.760 − 1.31i)17-s + (−0.925 − 1.60i)19-s + (−0.696 − 0.253i)20-s + (0.399 + 0.334i)22-s + (0.794 − 0.289i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.597 + 0.802i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.597 + 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.632298 - 1.25900i\)
\(L(\frac12)\) \(\approx\) \(0.632298 - 1.25900i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.135 + 0.765i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (1.82 + 1.52i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (-2.35 - 0.855i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (2.40 - 2.01i)T + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (-0.232 + 1.31i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-3.13 + 5.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.03 + 6.98i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.81 + 1.38i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-1.61 - 9.14i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (2.66 - 0.968i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (2.76 - 4.79i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.23 + 6.99i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-1.79 + 1.50i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (4.33 + 1.57i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + 0.135T + 53T^{2} \)
59 \( 1 + (-3.06 - 2.57i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (0.321 + 0.116i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-1.75 + 9.96i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-4.09 + 7.09i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.15 - 10.6i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.708 - 4.01i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (0.158 + 0.899i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (1.86 + 3.22i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.59 - 3.85i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.34414360119954009209318712915, −9.198954687134584910606758945917, −8.469178349751270241929313556184, −7.46472847739533431468441887057, −6.82217388554008466840426957736, −5.18641764574832030058359066016, −4.81827161688150518265058162356, −3.24930917477564873574310466927, −2.20666869344269542209821326743, −0.74309652909208492248116972486, 1.85413185377550065953813161562, 3.24627131995384667517310158913, 4.14733115677932738061804938173, 5.62407144568970486630070735507, 6.33035030295798733444331824558, 7.45514061986920784779674468079, 7.994993271353724637877885983946, 8.412525216572854827918291203702, 10.05259643704533610185017044444, 10.96698355773572549157141440921

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