Properties

Label 2-3e6-27.4-c1-0-30
Degree $2$
Conductor $729$
Sign $-0.597 + 0.802i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.162 + 0.0591i)2-s + (−1.50 − 1.26i)4-s + (0.648 − 3.67i)5-s + (2.32 − 1.94i)7-s + (−0.343 − 0.594i)8-s + (0.323 − 0.559i)10-s + (0.432 + 2.45i)11-s + (0.718 − 0.261i)13-s + (0.492 − 0.179i)14-s + (0.663 + 3.76i)16-s + (2.31 − 4.00i)17-s + (0.305 + 0.529i)19-s + (−5.63 + 4.73i)20-s + (−0.0748 + 0.424i)22-s + (−4.99 − 4.19i)23-s + ⋯
L(s)  = 1  + (0.114 + 0.0418i)2-s + (−0.754 − 0.633i)4-s + (0.290 − 1.64i)5-s + (0.877 − 0.736i)7-s + (−0.121 − 0.210i)8-s + (0.102 − 0.177i)10-s + (0.130 + 0.739i)11-s + (0.199 − 0.0725i)13-s + (0.131 − 0.0479i)14-s + (0.165 + 0.940i)16-s + (0.560 − 0.970i)17-s + (0.0701 + 0.121i)19-s + (−1.26 + 1.05i)20-s + (−0.0159 + 0.0904i)22-s + (−1.04 − 0.874i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.597 + 0.802i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (568, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.597 + 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.635538 - 1.26546i\)
\(L(\frac12)\) \(\approx\) \(0.635538 - 1.26546i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.162 - 0.0591i)T + (1.53 + 1.28i)T^{2} \)
5 \( 1 + (-0.648 + 3.67i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (-2.32 + 1.94i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (-0.432 - 2.45i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (-0.718 + 0.261i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-2.31 + 4.00i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.305 - 0.529i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.99 + 4.19i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (6.15 + 2.24i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-5.01 - 4.21i)T + (5.38 + 30.5i)T^{2} \)
37 \( 1 + (2.47 - 4.29i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.94 + 1.79i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-0.967 - 5.48i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (0.848 - 0.711i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + 8.84T + 53T^{2} \)
59 \( 1 + (-2.05 + 11.6i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-6.27 + 5.26i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-1.13 + 0.414i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (2.45 - 4.26i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.14 + 3.72i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-11.0 - 4.03i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-8.47 - 3.08i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-3.76 - 6.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.164 + 0.933i)T + (-91.1 + 33.1i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.786989932941252880062367250165, −9.422908562415906250739853224006, −8.357595972183247266069964997951, −7.80901383605697753008064005977, −6.35791255540972834967176495203, −5.19831446651974815691015006571, −4.77865720077543700945367009190, −4.00993054631011286708636346932, −1.73704447406699338304554622183, −0.76344460732520035621483390127, 2.08242087981993995256167566743, 3.23880577498883384389646832375, 4.00665366933738201892528074819, 5.51047213106314644449474847413, 6.09155102361642059268719458804, 7.41310641337891193073492370669, 8.046541089256525726709956234217, 8.926228885465578911052453951140, 9.866573880994295126513521044598, 10.78715736337647021221331748806

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