Properties

Label 2-3e6-243.151-c1-0-12
Degree $2$
Conductor $729$
Sign $0.774 - 0.632i$
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  + (0.129 − 0.0359i)2-s + (−1.69 + 1.02i)4-s + (2.51 + 0.0976i)5-s + (4.08 − 0.638i)7-s + (−0.366 + 0.388i)8-s + (0.328 − 0.0778i)10-s + (−0.654 + 4.78i)11-s + (−0.189 − 0.276i)13-s + (0.504 − 0.229i)14-s + (1.81 − 3.44i)16-s + (−1.56 − 0.787i)17-s + (0.123 − 2.12i)19-s + (−4.36 + 2.41i)20-s + (0.0877 + 0.642i)22-s + (0.620 + 1.60i)23-s + ⋯
L(s)  = 1  + (0.0913 − 0.0254i)2-s + (−0.848 + 0.512i)4-s + (1.12 + 0.0436i)5-s + (1.54 − 0.241i)7-s + (−0.129 + 0.137i)8-s + (0.103 − 0.0246i)10-s + (−0.197 + 1.44i)11-s + (−0.0526 − 0.0767i)13-s + (0.134 − 0.0613i)14-s + (0.453 − 0.860i)16-s + (−0.380 − 0.190i)17-s + (0.0283 − 0.487i)19-s + (−0.977 + 0.539i)20-s + (0.0186 + 0.136i)22-s + (0.129 + 0.335i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.774 - 0.632i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.774 - 0.632i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71859 + 0.612069i\)
\(L(\frac12)\) \(\approx\) \(1.71859 + 0.612069i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.129 + 0.0359i)T + (1.71 - 1.03i)T^{2} \)
5 \( 1 + (-2.51 - 0.0976i)T + (4.98 + 0.387i)T^{2} \)
7 \( 1 + (-4.08 + 0.638i)T + (6.66 - 2.13i)T^{2} \)
11 \( 1 + (0.654 - 4.78i)T + (-10.5 - 2.94i)T^{2} \)
13 \( 1 + (0.189 + 0.276i)T + (-4.68 + 12.1i)T^{2} \)
17 \( 1 + (1.56 + 0.787i)T + (10.1 + 13.6i)T^{2} \)
19 \( 1 + (-0.123 + 2.12i)T + (-18.8 - 2.20i)T^{2} \)
23 \( 1 + (-0.620 - 1.60i)T + (-17.0 + 15.4i)T^{2} \)
29 \( 1 + (-0.238 + 0.170i)T + (9.38 - 27.4i)T^{2} \)
31 \( 1 + (-5.27 - 6.04i)T + (-4.19 + 30.7i)T^{2} \)
37 \( 1 + (-6.18 + 8.31i)T + (-10.6 - 35.4i)T^{2} \)
41 \( 1 + (-1.90 - 7.40i)T + (-35.8 + 19.8i)T^{2} \)
43 \( 1 + (0.684 - 0.620i)T + (4.16 - 42.7i)T^{2} \)
47 \( 1 + (4.38 - 5.02i)T + (-6.36 - 46.5i)T^{2} \)
53 \( 1 + (-2.13 - 12.1i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (-1.52 - 0.623i)T + (42.1 + 41.3i)T^{2} \)
61 \( 1 + (0.475 + 0.287i)T + (28.4 + 53.9i)T^{2} \)
67 \( 1 + (-6.22 - 4.45i)T + (21.6 + 63.3i)T^{2} \)
71 \( 1 + (-3.03 + 10.1i)T + (-59.3 - 39.0i)T^{2} \)
73 \( 1 + (10.8 + 2.56i)T + (65.2 + 32.7i)T^{2} \)
79 \( 1 + (11.3 + 11.1i)T + (1.53 + 78.9i)T^{2} \)
83 \( 1 + (-3.20 + 12.4i)T + (-72.6 - 40.0i)T^{2} \)
89 \( 1 + (2.10 + 7.03i)T + (-74.3 + 48.9i)T^{2} \)
97 \( 1 + (-0.615 + 0.0238i)T + (96.7 - 7.51i)T^{2} \)
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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

−10.34384516610544995604965773426, −9.584635116728913271160218149471, −8.871200675633341326579994674156, −7.87260314373811646589038507867, −7.22306781207494882375331179824, −5.83708367643159024824743556239, −4.73839279605340879924598354195, −4.50665261944917320018404642480, −2.68778828665969101427304520475, −1.53510033750068317411054984549, 1.11068006733504424329624811257, 2.32352429063195538382966788947, 4.00400555150647635157867434122, 5.09405448190912865738837788852, 5.62919584862181432307008890293, 6.44936482129508131903783035868, 8.226527984912673572930521328114, 8.404196976016670643724120888904, 9.486326382223987648672467377861, 10.22404243816809609994246016874

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