Properties

Label 2-3e6-27.25-c1-0-27
Degree $2$
Conductor $729$
Sign $0.116 + 0.993i$
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  + (0.118 − 0.673i)2-s + (1.43 + 0.524i)4-s + (0.802 − 0.673i)5-s + (0.113 − 0.0412i)7-s + (1.20 − 2.09i)8-s + (−0.358 − 0.620i)10-s + (−4.16 − 3.49i)11-s + (−0.794 − 4.50i)13-s + (−0.0143 − 0.0812i)14-s + (1.08 + 0.907i)16-s + (−2.38 − 4.13i)17-s + (0.294 − 0.509i)19-s + (1.50 − 0.549i)20-s + (−2.84 + 2.38i)22-s + (7.32 + 2.66i)23-s + ⋯
L(s)  = 1  + (0.0839 − 0.476i)2-s + (0.719 + 0.262i)4-s + (0.359 − 0.301i)5-s + (0.0428 − 0.0155i)7-s + (0.427 − 0.739i)8-s + (−0.113 − 0.196i)10-s + (−1.25 − 1.05i)11-s + (−0.220 − 1.24i)13-s + (−0.00382 − 0.0217i)14-s + (0.270 + 0.226i)16-s + (−0.579 − 1.00i)17-s + (0.0675 − 0.116i)19-s + (0.337 − 0.122i)20-s + (−0.607 + 0.509i)22-s + (1.52 + 0.555i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.116 + 0.993i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.116 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42251 - 1.26593i\)
\(L(\frac12)\) \(\approx\) \(1.42251 - 1.26593i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.118 + 0.673i)T + (-1.87 - 0.684i)T^{2} \)
5 \( 1 + (-0.802 + 0.673i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (-0.113 + 0.0412i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (4.16 + 3.49i)T + (1.91 + 10.8i)T^{2} \)
13 \( 1 + (0.794 + 4.50i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (2.38 + 4.13i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.294 + 0.509i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-7.32 - 2.66i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.880 + 4.99i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-8.23 - 2.99i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (-1.09 - 1.89i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.31 - 7.44i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (1 + 0.839i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (2.27 - 0.826i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + 3.04T + 53T^{2} \)
59 \( 1 + (0.0336 - 0.0282i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-9.59 + 3.49i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.322 + 1.83i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-3.25 - 5.63i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.11 - 10.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.121 + 0.691i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-1.17 + 6.67i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (3.42 - 5.93i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.91 + 5.80i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.30944256413542174623594352015, −9.570146870777693592366127694154, −8.337127422122223696163700531666, −7.70506024202875819792116362094, −6.69697666311433462081808272432, −5.61311025050166861941500784449, −4.81807969641304416818076977460, −3.11320807913761258177775993431, −2.71350252133016895899776913971, −0.966454563550970335119922011037, 1.89084738641495041672478950555, 2.67015658190047962158852866762, 4.45599997355821570822878007332, 5.24733945253350000671158137674, 6.43963999550525961659308901203, 6.88912371615983400908499040975, 7.81784695899641347579127120371, 8.767870317041011134806052257765, 9.952968839739956487829124766597, 10.52490400663501758622133312427

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