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Properties

Label 2-3e6-27.22-c1-0-18
Degree $2$
Conductor $729$
Sign $0.993 + 0.116i$
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  + (1.20 − 1.01i)2-s + (0.0853 − 0.483i)4-s + (1.57 − 0.574i)5-s + (0.482 + 2.73i)7-s + (1.19 + 2.06i)8-s + (1.32 − 2.29i)10-s + (−3.90 − 1.41i)11-s + (5.26 + 4.41i)13-s + (3.36 + 2.81i)14-s + (4.45 + 1.62i)16-s + (0.488 − 0.845i)17-s + (−1.34 − 2.32i)19-s + (−0.143 − 0.812i)20-s + (−6.15 + 2.24i)22-s + (−0.280 + 1.58i)23-s + ⋯
L(s)  = 1  + (0.854 − 0.717i)2-s + (0.0426 − 0.241i)4-s + (0.705 − 0.256i)5-s + (0.182 + 1.03i)7-s + (0.420 + 0.729i)8-s + (0.418 − 0.725i)10-s + (−1.17 − 0.428i)11-s + (1.46 + 1.22i)13-s + (0.898 + 0.753i)14-s + (1.11 + 0.405i)16-s + (0.118 − 0.205i)17-s + (−0.308 − 0.533i)19-s + (−0.0320 − 0.181i)20-s + (−1.31 + 0.477i)22-s + (−0.0584 + 0.331i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.993 + 0.116i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.993 + 0.116i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.69653 - 0.157055i\)
\(L(\frac12)\) \(\approx\) \(2.69653 - 0.157055i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.20 + 1.01i)T + (0.347 - 1.96i)T^{2} \)
5 \( 1 + (-1.57 + 0.574i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (-0.482 - 2.73i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (3.90 + 1.41i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (-5.26 - 4.41i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.488 + 0.845i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.34 + 2.32i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.280 - 1.58i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-6.30 + 5.28i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-0.181 + 1.02i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (-0.654 + 1.13i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.71 - 3.11i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (9.24 + 3.36i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (2.17 + 12.3i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + 7.34T + 53T^{2} \)
59 \( 1 + (-8.50 + 3.09i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (0.223 + 1.26i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (3.55 + 2.98i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-2.81 + 4.87i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.28 - 3.95i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.56 + 2.99i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (4.41 - 3.70i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (-2.27 - 3.93i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.05 + 2.93i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.64951790727712446526183225068, −9.586663975902543337051108741323, −8.629906550504637773620123745772, −8.095519029831740829739320770718, −6.50681592971676960649775989423, −5.61388287960138199912892839790, −4.95717705187961298713356025618, −3.79803093107750623640888962079, −2.63851372297280956250664876446, −1.80829686228648276669424081989, 1.24268772299173432549447153068, 3.04648323457374325379816934716, 4.15492682314594236007902950467, 5.12586927324043773552328793465, 5.95264552498003609574194772098, 6.62892563385373133566273416650, 7.68287039531845740402759558385, 8.329447247023790342705912743989, 9.906168734344787103045661698861, 10.43071038208761870896436438286

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