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Properties

Label 2-3e6-27.4-c1-0-15
Degree $2$
Conductor $729$
Sign $-0.396 - 0.918i$
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  + (2.30 + 0.840i)2-s + (3.09 + 2.59i)4-s + (−0.534 + 3.03i)5-s + (−2.03 + 1.70i)7-s + (2.50 + 4.34i)8-s + (−3.78 + 6.55i)10-s + (−0.596 − 3.38i)11-s + (3.14 − 1.14i)13-s + (−6.14 + 2.23i)14-s + (0.737 + 4.18i)16-s + (−1.28 + 2.22i)17-s + (1.04 + 1.81i)19-s + (−9.53 + 8.00i)20-s + (1.46 − 8.31i)22-s + (0.409 + 0.343i)23-s + ⋯
L(s)  = 1  + (1.63 + 0.594i)2-s + (1.54 + 1.29i)4-s + (−0.239 + 1.35i)5-s + (−0.769 + 0.645i)7-s + (0.886 + 1.53i)8-s + (−1.19 + 2.07i)10-s + (−0.179 − 1.01i)11-s + (0.871 − 0.317i)13-s + (−1.64 + 0.597i)14-s + (0.184 + 1.04i)16-s + (−0.311 + 0.540i)17-s + (0.240 + 0.416i)19-s + (−2.13 + 1.78i)20-s + (0.312 − 1.77i)22-s + (0.0853 + 0.0716i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.396 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.396 - 0.918i$
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: \((2,\ 729,\ (\ :1/2),\ -0.396 - 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86699 + 2.83862i\)
\(L(\frac12)\) \(\approx\) \(1.86699 + 2.83862i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.30 - 0.840i)T + (1.53 + 1.28i)T^{2} \)
5 \( 1 + (0.534 - 3.03i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (2.03 - 1.70i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (0.596 + 3.38i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (-3.14 + 1.14i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (1.28 - 2.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.04 - 1.81i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.409 - 0.343i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (2.37 + 0.865i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-5.90 - 4.95i)T + (5.38 + 30.5i)T^{2} \)
37 \( 1 + (-5.14 + 8.91i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.59 + 1.67i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-0.476 - 2.69i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (4.33 - 3.63i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 - 6.42T + 53T^{2} \)
59 \( 1 + (-0.287 + 1.62i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-11.0 + 9.23i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-5.52 + 2.01i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (7.40 - 12.8i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.940 + 1.62i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (16.1 + 5.88i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (3.72 + 1.35i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (2.54 + 4.41i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.84 + 10.4i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.10427294390647441141836601331, −10.05370503652485037045224717187, −8.682311408835488317179316617326, −7.69508256258881434901298929969, −6.74791174098436459835219963072, −6.08026284637269566268208451922, −5.62551052155399624812049878389, −4.06201790894318646344144202363, −3.27798184517242948414552330028, −2.67001699657231865237074776980, 1.12521454920176276856875776862, 2.59286147251713898150236462377, 3.89536409788632413610039585663, 4.46635942251592782340556361941, 5.23886880141667638794593675707, 6.31713320066779350035000020576, 7.16677703170933462815771026144, 8.450421202670083845388169512110, 9.526279040313299854924465546681, 10.26415585638854757140507180466

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