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Properties

Label 2-3e6-9.4-c1-0-4
Degree $2$
Conductor $729$
Sign $1$
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  + (−1.20 − 2.07i)2-s + (−1.88 + 3.26i)4-s + (0.0465 − 0.0806i)5-s + (0.289 + 0.502i)7-s + 4.24·8-s − 0.223·10-s + (−1.54 − 2.67i)11-s + (−2.10 + 3.63i)13-s + (0.696 − 1.20i)14-s + (−1.33 − 2.30i)16-s + 1.99·17-s − 3.84·19-s + (0.175 + 0.303i)20-s + (−3.71 + 6.43i)22-s + (−2.22 + 3.85i)23-s + ⋯
L(s)  = 1  + (−0.849 − 1.47i)2-s + (−0.941 + 1.63i)4-s + (0.0208 − 0.0360i)5-s + (0.109 + 0.189i)7-s + 1.50·8-s − 0.0706·10-s + (−0.466 − 0.807i)11-s + (−0.582 + 1.00i)13-s + (0.186 − 0.322i)14-s + (−0.332 − 0.576i)16-s + 0.482·17-s − 0.882·19-s + (0.0392 + 0.0679i)20-s + (−0.791 + 1.37i)22-s + (−0.464 + 0.804i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $1$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.603867\)
\(L(\frac12)\) \(\approx\) \(0.603867\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.20 + 2.07i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.0465 + 0.0806i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.289 - 0.502i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.54 + 2.67i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.10 - 3.63i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.99T + 17T^{2} \)
19 \( 1 + 3.84T + 19T^{2} \)
23 \( 1 + (2.22 - 3.85i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.19 - 5.54i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.828 - 1.43i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.03T + 37T^{2} \)
41 \( 1 + (0.548 - 0.949i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.45 - 5.97i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.79 - 3.11i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 5.40T + 53T^{2} \)
59 \( 1 + (5.14 - 8.90i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.59 - 11.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.41 + 7.65i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.14T + 71T^{2} \)
73 \( 1 - 0.195T + 73T^{2} \)
79 \( 1 + (-3.60 - 6.24i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.45 + 12.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 1.55T + 89T^{2} \)
97 \( 1 + (2.64 + 4.58i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.48495290685106525914553824753, −9.601705930137132532754475457162, −8.937948987531248722008983797847, −8.216649055259409269275467491328, −7.22068833088997890154844146105, −5.88009514516183206321319539418, −4.60894590856397283668272401636, −3.46146775822478352731439427526, −2.50251305211539350673646755801, −1.31923929196191333655543787689, 0.44545804738154694529083231860, 2.47750109491600768040023954805, 4.35874085488898620415718210678, 5.27192762072521682469495561397, 6.20811320682108828695295725966, 7.00434444644553379930971191220, 7.949521347427938508861972190179, 8.250221282168666201872279093508, 9.459259842493886844688501950404, 10.12831450288122739485537844272

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