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Properties

Label 2-3e6-27.25-c1-0-25
Degree $2$
Conductor $729$
Sign $-0.0581 + 0.998i$
Analytic cond. $5.82109$
Root an. cond. $2.41269$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.233 + 1.32i)2-s + (0.173 + 0.0632i)4-s + (−1.26 + 1.06i)5-s + (−2.26 + 0.824i)7-s + (−1.47 + 2.54i)8-s + (−1.11 − 1.92i)10-s + (−4.55 − 3.82i)11-s + (−0.560 − 3.17i)13-s + (−0.564 − 3.19i)14-s + (−2.75 − 2.31i)16-s + (1.5 + 2.59i)17-s + (3.31 − 5.74i)19-s + (−0.286 + 0.104i)20-s + (6.13 − 5.14i)22-s + (−2.76 − 1.00i)23-s + ⋯
L(s)  = 1  + (−0.165 + 0.938i)2-s + (0.0868 + 0.0316i)4-s + (−0.566 + 0.475i)5-s + (−0.856 + 0.311i)7-s + (−0.520 + 0.901i)8-s + (−0.352 − 0.609i)10-s + (−1.37 − 1.15i)11-s + (−0.155 − 0.881i)13-s + (−0.150 − 0.855i)14-s + (−0.688 − 0.577i)16-s + (0.363 + 0.630i)17-s + (0.761 − 1.31i)19-s + (−0.0641 + 0.0233i)20-s + (1.30 − 1.09i)22-s + (−0.576 − 0.209i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0581 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0581 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.0581 + 0.998i$
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: \(1\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.0581 + 0.998i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.233 - 1.32i)T + (-1.87 - 0.684i)T^{2} \)
5 \( 1 + (1.26 - 1.06i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (2.26 - 0.824i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (4.55 + 3.82i)T + (1.91 + 10.8i)T^{2} \)
13 \( 1 + (0.560 + 3.17i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.31 + 5.74i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.76 + 1.00i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.224 + 1.27i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-0.553 - 0.201i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (0.0209 + 0.0362i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.851 - 4.82i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (3.97 + 3.33i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (3.51 - 1.27i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + 11.6T + 53T^{2} \)
59 \( 1 + (-5.62 + 4.72i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (10.3 - 3.77i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-0.322 - 1.82i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (2.75 + 4.77i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.77 - 4.81i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.656 - 3.72i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (0.692 - 3.92i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (-4.07 + 7.05i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.199 - 0.167i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.19224901590749057625432129810, −9.085388446382290060630927888392, −8.046817099778823132130966232470, −7.72825258835332031010964991155, −6.65909991892793856715981727232, −5.88373849714584139387447137164, −5.12345822805052969623259889195, −3.24701946461581273194775912274, −2.81149353380897269419651915773, 0, 1.72180626619681689135093834528, 2.92882832440759430249519313068, 3.92262555841052825930223146402, 4.99305168930181618231918895683, 6.26884842238746387175008116557, 7.27805549780888588140392167323, 7.958361677873360754977437551602, 9.331038634664128067616212169546, 9.942255958931019922032522433849

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