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Properties

Label 2-3e6-9.7-c1-0-27
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: $\chi_{729} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.05396i\)
\(L(\frac12)\) \(\approx\) \(2.05396i\)
\(L(\frac{3}{2})\) 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

−10.38013119863522158601185311129, −9.393627122089562649873102284711, −8.526178573095831422889819625084, −7.31017213147225546360151138857, −6.01962905505180950279738196404, −5.14658140702591933117742983272, −4.18430871805490946191751547984, −3.28088289268776676325307816717, −2.28348785776799466059598986503, −0.826329777473242051696321746943, 2.24405101095848888143818647200, 3.96052035652500293117992375025, 4.57412923632367223297153416956, 5.48296813405402929163362496133, 6.66833084951768383387787414857, 6.92000462430238859598252928512, 7.985158943255732436026250713673, 8.870356636758902610880890935758, 9.639527590504352397338237042936, 10.94521772996242322701957779766

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