Properties

Label 2-3e6-27.13-c1-0-5
Degree $2$
Conductor $729$
Sign $-0.802 - 0.597i$
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  + (0.342 + 1.93i)2-s + (−1.76 + 0.642i)4-s + (−2.83 − 2.37i)5-s + (2.20 + 0.802i)7-s + (0.118 + 0.205i)8-s + (3.64 − 6.31i)10-s + (1.66 − 1.40i)11-s + (−0.819 + 4.64i)13-s + (−0.802 + 4.55i)14-s + (−3.23 + 2.71i)16-s + (−1.46 + 2.54i)17-s + (3.11 + 5.39i)19-s + (6.53 + 2.37i)20-s + (3.28 + 2.75i)22-s + (0.487 − 0.177i)23-s + ⋯
L(s)  = 1  + (0.241 + 1.37i)2-s + (−0.883 + 0.321i)4-s + (−1.26 − 1.06i)5-s + (0.833 + 0.303i)7-s + (0.0419 + 0.0727i)8-s + (1.15 − 1.99i)10-s + (0.503 − 0.422i)11-s + (−0.227 + 1.28i)13-s + (−0.214 + 1.21i)14-s + (−0.809 + 0.679i)16-s + (−0.355 + 0.616i)17-s + (0.714 + 1.23i)19-s + (1.46 + 0.532i)20-s + (0.700 + 0.588i)22-s + (0.101 − 0.0370i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.802 - 0.597i$
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: \((2,\ 729,\ (\ :1/2),\ -0.802 - 0.597i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.440816 + 1.33030i\)
\(L(\frac12)\) \(\approx\) \(0.440816 + 1.33030i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.342 - 1.93i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (2.83 + 2.37i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (-2.20 - 0.802i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (-1.66 + 1.40i)T + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (0.819 - 4.64i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (1.46 - 2.54i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.11 - 5.39i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.487 + 0.177i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.606 - 3.43i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-4.04 + 1.47i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (1.20 - 2.08i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.433 - 2.46i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-0.815 + 0.684i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (0.223 + 0.0812i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + 4.66T + 53T^{2} \)
59 \( 1 + (-10.1 - 8.55i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-3.45 - 1.25i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-2.48 + 14.0i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-0.601 + 1.04i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.34 - 4.05i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.22 + 12.6i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-1.96 - 11.1i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (-0.349 - 0.605i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.42 - 4.55i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.98729065307781120066598388437, −9.411367434393645349364378989381, −8.464373834301224561913012532371, −8.230686836802652685231231113375, −7.32367231144121704617291531504, −6.38955539869917007487464383840, −5.31685190516264751071033254262, −4.56885130437116771512595321411, −3.85064078668689623597275652729, −1.56813188724376476017562289276, 0.72874453893939629382222705315, 2.48135946173634014825747005873, 3.26464688356820108614115600272, 4.21009609541756222484170042031, 5.03242883489480983622998201489, 6.85247708306732900166872628448, 7.40489216601895701935395882338, 8.296555476105740883894554645120, 9.598523565755410713613100213445, 10.39339799275194551371709484078

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