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Properties

Label 2-3e6-9.4-c1-0-20
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  + (0.388 + 0.673i)2-s + (0.697 − 1.20i)4-s + (1.18 − 2.05i)5-s + (1.25 + 2.16i)7-s + 2.64·8-s + 1.84·10-s + (1.57 + 2.72i)11-s + (−0.668 + 1.15i)13-s + (−0.972 + 1.68i)14-s + (−0.368 − 0.637i)16-s − 6.27·17-s + 8.06·19-s + (−1.65 − 2.87i)20-s + (−1.22 + 2.11i)22-s + (2.02 − 3.51i)23-s + ⋯
L(s)  = 1  + (0.274 + 0.476i)2-s + (0.348 − 0.604i)4-s + (0.531 − 0.920i)5-s + (0.472 + 0.818i)7-s + 0.933·8-s + 0.584·10-s + (0.473 + 0.820i)11-s + (−0.185 + 0.321i)13-s + (−0.259 + 0.450i)14-s + (−0.0920 − 0.159i)16-s − 1.52·17-s + 1.85·19-s + (−0.370 − 0.642i)20-s + (−0.260 + 0.451i)22-s + (0.422 − 0.732i)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

\(L(1)\) \(\approx\) \(2.32882\)
\(L(\frac12)\) \(\approx\) \(2.32882\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.388 - 0.673i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.18 + 2.05i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.25 - 2.16i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.57 - 2.72i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.668 - 1.15i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 6.27T + 17T^{2} \)
19 \( 1 - 8.06T + 19T^{2} \)
23 \( 1 + (-2.02 + 3.51i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.64 + 8.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.41 - 2.45i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.53T + 37T^{2} \)
41 \( 1 + (3.55 - 6.15i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.16 + 2.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.30 + 3.99i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.135T + 53T^{2} \)
59 \( 1 + (1.99 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.170 + 0.296i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.06 - 8.76i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.19T + 71T^{2} \)
73 \( 1 + 12.3T + 73T^{2} \)
79 \( 1 + (2.04 + 3.53i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.456 - 0.790i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3.72T + 89T^{2} \)
97 \( 1 + (-2.99 - 5.19i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.22374936551372005808025845885, −9.390575623874343473979726831055, −8.879496355611182204470469227716, −7.64751552286770437342862948403, −6.76374724687261232175710179859, −5.82807747654853387927488908603, −5.04553970700473545110208740260, −4.43231052278617749852646212338, −2.35949029598725030653252059206, −1.41892294634490554588762506530, 1.53965201446149950478751002517, 2.88879389064029771475394957812, 3.58827201348560494321467195076, 4.76254488527853835085174320210, 6.01708455301529318495399845725, 7.14473376277541375493269445153, 7.45439018504141444559277047007, 8.715031845666089433869888043123, 9.712141844987246544041214406452, 10.82714480498227681716694696944

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