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Properties

Label 2-3e6-9.4-c1-0-23
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  + (1.06 + 1.84i)2-s + (−1.25 + 2.17i)4-s + (1.03 − 1.79i)5-s + (−2.42 − 4.19i)7-s − 1.09·8-s + 4.40·10-s + (−2.07 − 3.59i)11-s + (0.608 − 1.05i)13-s + (5.15 − 8.92i)14-s + (1.35 + 2.34i)16-s + 2.36·17-s − 1.83·19-s + (2.60 + 4.51i)20-s + (4.40 − 7.63i)22-s + (2.15 − 3.72i)23-s + ⋯
L(s)  = 1  + (0.751 + 1.30i)2-s + (−0.628 + 1.08i)4-s + (0.463 − 0.802i)5-s + (−0.916 − 1.58i)7-s − 0.387·8-s + 1.39·10-s + (−0.625 − 1.08i)11-s + (0.168 − 0.292i)13-s + (1.37 − 2.38i)14-s + (0.337 + 0.585i)16-s + 0.573·17-s − 0.421·19-s + (0.582 + 1.00i)20-s + (0.939 − 1.62i)22-s + (0.448 − 0.777i)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

\(L(1)\) \(\approx\) \(2.05881\)
\(L(\frac12)\) \(\approx\) \(2.05881\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.06 - 1.84i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.03 + 1.79i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.42 + 4.19i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.07 + 3.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.608 + 1.05i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2.36T + 17T^{2} \)
19 \( 1 + 1.83T + 19T^{2} \)
23 \( 1 + (-2.15 + 3.72i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.49 + 2.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.736 - 1.27i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.97T + 37T^{2} \)
41 \( 1 + (1.13 - 1.95i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.74 - 4.75i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.59 + 6.22i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6.32T + 53T^{2} \)
59 \( 1 + (0.131 - 0.227i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.22 - 3.85i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.06 - 3.57i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.08T + 71T^{2} \)
73 \( 1 - 12.7T + 73T^{2} \)
79 \( 1 + (2.27 + 3.94i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.22 - 7.32i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 16.9T + 89T^{2} \)
97 \( 1 + (-2.55 - 4.42i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.36488025849131436839282840170, −9.432631514221575980545528509190, −8.295774499804542427753772962716, −7.67637871918699686789550025777, −6.67257899514189871944796635446, −6.02132035548371504795507392952, −5.12339211448574239808331945169, −4.19966145721122117467101804998, −3.23095991775238003884677667742, −0.852842285448942225754693816618, 2.00563436551980738890601090915, 2.65419853693720718136540110051, 3.48419722699261839375006678852, 4.85646632121062546524893081114, 5.71814502811794881851596647262, 6.59920022801620914766740892804, 7.77579868333630303906759406367, 9.252568888996885186615510028668, 9.703215494138347284855120359585, 10.51279866547905573869183452325

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