Properties

Label 2-3e6-81.22-c1-0-12
Degree $2$
Conductor $729$
Sign $-0.294 + 0.955i$
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  + (−1.31 − 1.39i)2-s + (−0.0981 + 1.68i)4-s + (−0.783 + 0.0915i)5-s + (1.06 + 0.534i)7-s + (−0.459 + 0.385i)8-s + (1.15 + 0.973i)10-s + (−0.814 − 1.88i)11-s + (2.96 + 0.703i)13-s + (−0.656 − 2.19i)14-s + (4.49 + 0.525i)16-s + (−0.788 + 4.47i)17-s + (−0.838 − 4.75i)19-s + (−0.0773 − 1.32i)20-s + (−1.56 + 3.62i)22-s + (5.89 − 2.95i)23-s + ⋯
L(s)  = 1  + (−0.931 − 0.987i)2-s + (−0.0490 + 0.842i)4-s + (−0.350 + 0.0409i)5-s + (0.402 + 0.202i)7-s + (−0.162 + 0.136i)8-s + (0.366 + 0.307i)10-s + (−0.245 − 0.569i)11-s + (0.823 + 0.195i)13-s + (−0.175 − 0.585i)14-s + (1.12 + 0.131i)16-s + (−0.191 + 1.08i)17-s + (−0.192 − 1.09i)19-s + (−0.0173 − 0.297i)20-s + (−0.333 + 0.773i)22-s + (1.22 − 0.617i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.294 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.294 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.294 + 0.955i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.294 + 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.493671 - 0.668355i\)
\(L(\frac12)\) \(\approx\) \(0.493671 - 0.668355i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.31 + 1.39i)T + (-0.116 + 1.99i)T^{2} \)
5 \( 1 + (0.783 - 0.0915i)T + (4.86 - 1.15i)T^{2} \)
7 \( 1 + (-1.06 - 0.534i)T + (4.18 + 5.61i)T^{2} \)
11 \( 1 + (0.814 + 1.88i)T + (-7.54 + 8.00i)T^{2} \)
13 \( 1 + (-2.96 - 0.703i)T + (11.6 + 5.83i)T^{2} \)
17 \( 1 + (0.788 - 4.47i)T + (-15.9 - 5.81i)T^{2} \)
19 \( 1 + (0.838 + 4.75i)T + (-17.8 + 6.49i)T^{2} \)
23 \( 1 + (-5.89 + 2.95i)T + (13.7 - 18.4i)T^{2} \)
29 \( 1 + (-2.88 + 9.63i)T + (-24.2 - 15.9i)T^{2} \)
31 \( 1 + (-7.44 - 4.89i)T + (12.2 + 28.4i)T^{2} \)
37 \( 1 + (0.563 - 0.205i)T + (28.3 - 23.7i)T^{2} \)
41 \( 1 + (-0.763 + 0.808i)T + (-2.38 - 40.9i)T^{2} \)
43 \( 1 + (-4.47 + 6.00i)T + (-12.3 - 41.1i)T^{2} \)
47 \( 1 + (-0.765 + 0.503i)T + (18.6 - 43.1i)T^{2} \)
53 \( 1 + (-3.63 + 6.29i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.06 - 4.78i)T + (-40.4 - 42.9i)T^{2} \)
61 \( 1 + (-0.00126 - 0.0216i)T + (-60.5 + 7.08i)T^{2} \)
67 \( 1 + (4.31 + 14.4i)T + (-55.9 + 36.8i)T^{2} \)
71 \( 1 + (-3.88 - 3.26i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (0.586 - 0.492i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (5.44 + 5.77i)T + (-4.59 + 78.8i)T^{2} \)
83 \( 1 + (-9.18 - 9.73i)T + (-4.82 + 82.8i)T^{2} \)
89 \( 1 + (4.47 - 3.75i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (7.21 + 0.843i)T + (94.3 + 22.3i)T^{2} \)
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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

−10.33420590260078142684280677530, −9.276913200755932383231379066836, −8.476995987171749503534265719178, −8.126072324727659551841227344751, −6.69295770617867660961295541549, −5.70702628541793736221678272944, −4.39291894910617813208402746002, −3.19964370461765257368280658412, −2.11868155886137927376797154934, −0.72544093830497401023324151573, 1.14692775120099653587136120615, 3.08276583558687714942329870203, 4.39315935849028909949811083740, 5.53242463634759252427237814853, 6.51405564513200825820656395924, 7.41395710501976814054040430842, 7.941347058200834170033799435207, 8.777663950767989756648943973286, 9.557930480776205497888276223902, 10.40282644928083831943336167414

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