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Properties

Label 2-3e6-9.4-c1-0-21
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  + (−0.984 − 1.70i)2-s + (−0.939 + 1.62i)4-s + (1.85 − 3.20i)5-s + (1.17 + 2.03i)7-s − 0.237·8-s − 7.29·10-s + (−1.08 − 1.88i)11-s + (2.35 − 4.08i)13-s + (2.31 − 4.00i)14-s + (2.11 + 3.66i)16-s + 2.93·17-s − 6.22·19-s + (3.47 + 6.02i)20-s + (−2.14 + 3.71i)22-s + (0.259 − 0.449i)23-s + ⋯
L(s)  = 1  + (−0.696 − 1.20i)2-s + (−0.469 + 0.813i)4-s + (0.827 − 1.43i)5-s + (0.443 + 0.768i)7-s − 0.0839·8-s − 2.30·10-s + (−0.328 − 0.568i)11-s + (0.654 − 1.13i)13-s + (0.617 − 1.07i)14-s + (0.528 + 0.915i)16-s + 0.711·17-s − 1.42·19-s + (0.777 + 1.34i)20-s + (−0.457 + 0.792i)22-s + (0.0541 − 0.0937i)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

\(L(1)\) \(\approx\) \(1.09970i\)
\(L(\frac12)\) \(\approx\) \(1.09970i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.984 + 1.70i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.85 + 3.20i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.17 - 2.03i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.08 + 1.88i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.35 + 4.08i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2.93T + 17T^{2} \)
19 \( 1 + 6.22T + 19T^{2} \)
23 \( 1 + (-0.259 + 0.449i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.74 + 3.02i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.15 + 3.72i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.41T + 37T^{2} \)
41 \( 1 + (-1.24 + 2.16i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.532 + 0.921i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.118 + 0.205i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 4.66T + 53T^{2} \)
59 \( 1 + (6.65 - 11.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.83 - 3.18i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.14 - 12.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.20T + 71T^{2} \)
73 \( 1 + 4.68T + 73T^{2} \)
79 \( 1 + (-6.40 - 11.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.65 + 9.78i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 0.699T + 89T^{2} \)
97 \( 1 + (-3.54 - 6.13i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.02946403472072155405919370059, −9.121994724563930969890079568869, −8.539292526321715990636444872386, −8.061097215311714611062430662636, −5.88799114522623059189722411254, −5.66270878049593023214333996607, −4.30042438692821367574023599096, −2.84274357397700214846368122985, −1.82545267961753784815488932767, −0.72657846915495905354737958519, 1.83876358888238986358815443859, 3.27315380605514617259578191652, 4.68783395964814139339494249893, 6.05395578117833403698671848018, 6.56033652583126607303429966780, 7.25297311019405750819793495838, 7.991951455583657035487670697358, 9.056670494994224602220298480244, 9.860869380236037278176401275307, 10.60683512157953762822604188156

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