Properties

Label 2-3e6-27.4-c1-0-25
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.730 + 0.266i)2-s + (−1.06 − 0.896i)4-s + (−0.412 + 2.34i)5-s + (−1.91 + 1.60i)7-s + (−1.32 − 2.28i)8-s + (−0.924 + 1.60i)10-s + (−0.545 − 3.09i)11-s + (−1.25 + 0.457i)13-s + (−1.82 + 0.665i)14-s + (0.127 + 0.725i)16-s + (3.13 − 5.43i)17-s + (−4.03 − 6.98i)19-s + (2.53 − 2.13i)20-s + (0.424 − 2.40i)22-s + (−3.10 − 2.60i)23-s + ⋯
L(s)  = 1  + (0.516 + 0.188i)2-s + (−0.534 − 0.448i)4-s + (−0.184 + 1.04i)5-s + (−0.724 + 0.607i)7-s + (−0.466 − 0.808i)8-s + (−0.292 + 0.506i)10-s + (−0.164 − 0.932i)11-s + (−0.348 + 0.126i)13-s + (−0.488 + 0.177i)14-s + (0.0319 + 0.181i)16-s + (0.760 − 1.31i)17-s + (−0.925 − 1.60i)19-s + (0.567 − 0.476i)20-s + (0.0904 − 0.512i)22-s + (−0.647 − 0.543i)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

\(L(1)\) \(\approx\) \(0.216006 - 0.430103i\)
\(L(\frac12)\) \(\approx\) \(0.216006 - 0.430103i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.730 - 0.266i)T + (1.53 + 1.28i)T^{2} \)
5 \( 1 + (0.412 - 2.34i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (1.91 - 1.60i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (0.545 + 3.09i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (1.25 - 0.457i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-3.13 + 5.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.03 + 6.98i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.10 + 2.60i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (8.72 + 3.17i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-2.16 - 1.82i)T + (5.38 + 30.5i)T^{2} \)
37 \( 1 + (2.76 - 4.79i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.67 - 2.43i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-0.405 - 2.30i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-3.53 + 2.96i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + 0.135T + 53T^{2} \)
59 \( 1 + (-0.694 + 3.93i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-0.261 + 0.219i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (9.51 - 3.46i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-4.09 + 7.09i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.15 - 10.6i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.83 + 1.39i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-0.858 - 0.312i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (1.86 + 3.22i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.04 + 5.90i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.02174934031569449172020849922, −9.345055344016255946675315491366, −8.491276627361428042047997152842, −7.14740073862931848190060121442, −6.47427185993012610800712513389, −5.65769472827776126163382265337, −4.69516601661225939979425052622, −3.40550583553992846668692452740, −2.66932002829509761602312119378, −0.20384229118172381355196486345, 1.84116869113671053804583882412, 3.66703944305784590563584879475, 4.03456727242128250453172761060, 5.16372928594026477071824056352, 5.99544616257096618442643450494, 7.43475121912657064111121211044, 8.106719022479770449322854824862, 8.955211628925385073382777976378, 9.876491273837164823636877251256, 10.52999338515576202984355530164

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