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Properties

Label 2-3e6-9.7-c1-0-13
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: $\chi_{729} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.535249\)
\(L(\frac12)\) \(\approx\) \(0.535249\)
\(L(\frac{3}{2})\) 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

−9.868401371028462071416150527083, −9.062881274745307957290512233194, −8.645978490830257782600395183818, −8.134234329876681190615490924062, −6.72353934848757070520486989993, −6.15048710348385393118248210216, −5.46916428547636436435598129909, −4.11869877639348603431642193830, −2.63038763361332788757034121486, −0.41377968883626438970417442863, 1.15042868295493676050340453188, 2.65357760676528232762852293544, 3.68178327341010548873269318544, 4.25328526590342178884735285562, 6.29281645361653806465164054116, 7.10339606973433615431088800724, 7.78809170434363581614664555055, 9.129481977078202969815881591821, 9.781029129285543846674080160120, 10.46697662460315926745750134671

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