Properties

Label 2-3e6-27.13-c1-0-11
Degree $2$
Conductor $729$
Sign $0.116 - 0.993i$
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  + (0.223 + 1.26i)2-s + (0.326 − 0.118i)4-s + (0.342 + 0.286i)5-s + (3.31 + 1.20i)7-s + (1.50 + 2.61i)8-s + (−0.286 + 0.497i)10-s + (−2.12 + 1.78i)11-s + (0.571 − 3.24i)13-s + (−0.788 + 4.47i)14-s + (−2.43 + 2.04i)16-s + (3.51 − 6.09i)17-s + (2.59 + 4.49i)19-s + (0.145 + 0.0530i)20-s + (−2.73 − 2.29i)22-s + (−6.83 + 2.48i)23-s + ⋯
L(s)  = 1  + (0.157 + 0.895i)2-s + (0.163 − 0.0593i)4-s + (0.152 + 0.128i)5-s + (1.25 + 0.456i)7-s + (0.533 + 0.923i)8-s + (−0.0907 + 0.157i)10-s + (−0.642 + 0.538i)11-s + (0.158 − 0.898i)13-s + (−0.210 + 1.19i)14-s + (−0.609 + 0.511i)16-s + (0.853 − 1.47i)17-s + (0.594 + 1.03i)19-s + (0.0325 + 0.0118i)20-s + (−0.583 − 0.489i)22-s + (−1.42 + 0.518i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.116 - 0.993i$
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: \((2,\ 729,\ (\ :1/2),\ 0.116 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64048 + 1.45990i\)
\(L(\frac12)\) \(\approx\) \(1.64048 + 1.45990i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.223 - 1.26i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (-0.342 - 0.286i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (-3.31 - 1.20i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (2.12 - 1.78i)T + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (-0.571 + 3.24i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-3.51 + 6.09i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.59 - 4.49i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.83 - 2.48i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.628 - 3.56i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-1.81 + 0.662i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-1.61 + 2.79i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.844 - 4.78i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (4.41 - 3.70i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (2.83 + 1.03i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 - 8.77T + 53T^{2} \)
59 \( 1 + (-2.27 - 1.90i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (7.41 + 2.69i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (1.63 - 9.29i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (2.65 - 4.59i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.777 - 1.34i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.06 + 11.7i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (2.82 + 16.0i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (9.21 + 15.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.74 + 6.50i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.42204284882412835811045848123, −9.903594581397341581622148746310, −8.429559932289503579359023539947, −7.82476387687152806052289503713, −7.33367556318084413842246288322, −5.94706746313735144991734371450, −5.41972235232899322299572596489, −4.62212709714873164596355649478, −2.87006991808792782396662685596, −1.68886000283347128355313421753, 1.28031084662150670435868966997, 2.23383950060896805318108107847, 3.60440898636823535859337145282, 4.43779963788571805449049321832, 5.55987677790874399985789482485, 6.71966552652010550501180642994, 7.77682128830913304920251701123, 8.360261654486597762108794806131, 9.628046613860613007356083133450, 10.48765668331469789785800989253

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