Properties

Label 2-3e6-81.67-c1-0-28
Degree $2$
Conductor $729$
Sign $0.280 + 0.959i$
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  + (1.76 − 1.16i)2-s + (0.978 − 2.26i)4-s + (2.05 − 2.17i)5-s + (3.48 + 0.407i)7-s + (−0.172 − 0.977i)8-s + (1.09 − 6.22i)10-s + (−0.562 + 1.87i)11-s + (−0.0969 + 1.66i)13-s + (6.63 − 3.33i)14-s + (1.94 + 2.06i)16-s + (−3.65 + 1.33i)17-s + (−0.0155 − 0.00566i)19-s + (−2.92 − 6.77i)20-s + (1.18 + 3.97i)22-s + (−7.67 + 0.896i)23-s + ⋯
L(s)  = 1  + (1.24 − 0.821i)2-s + (0.489 − 1.13i)4-s + (0.917 − 0.972i)5-s + (1.31 + 0.154i)7-s + (−0.0609 − 0.345i)8-s + (0.347 − 1.96i)10-s + (−0.169 + 0.566i)11-s + (−0.0268 + 0.461i)13-s + (1.77 − 0.890i)14-s + (0.487 + 0.516i)16-s + (−0.886 + 0.322i)17-s + (−0.00357 − 0.00130i)19-s + (−0.653 − 1.51i)20-s + (0.253 + 0.846i)22-s + (−1.59 + 0.186i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.280 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.280 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.280 + 0.959i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.280 + 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.90903 - 2.18055i\)
\(L(\frac12)\) \(\approx\) \(2.90903 - 2.18055i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.76 + 1.16i)T + (0.792 - 1.83i)T^{2} \)
5 \( 1 + (-2.05 + 2.17i)T + (-0.290 - 4.99i)T^{2} \)
7 \( 1 + (-3.48 - 0.407i)T + (6.81 + 1.61i)T^{2} \)
11 \( 1 + (0.562 - 1.87i)T + (-9.19 - 6.04i)T^{2} \)
13 \( 1 + (0.0969 - 1.66i)T + (-12.9 - 1.50i)T^{2} \)
17 \( 1 + (3.65 - 1.33i)T + (13.0 - 10.9i)T^{2} \)
19 \( 1 + (0.0155 + 0.00566i)T + (14.5 + 12.2i)T^{2} \)
23 \( 1 + (7.67 - 0.896i)T + (22.3 - 5.30i)T^{2} \)
29 \( 1 + (4.12 + 2.07i)T + (17.3 + 23.2i)T^{2} \)
31 \( 1 + (6.12 + 8.23i)T + (-8.89 + 29.6i)T^{2} \)
37 \( 1 + (-5.48 - 4.60i)T + (6.42 + 36.4i)T^{2} \)
41 \( 1 + (4.72 + 3.10i)T + (16.2 + 37.6i)T^{2} \)
43 \( 1 + (4.44 - 1.05i)T + (38.4 - 19.2i)T^{2} \)
47 \( 1 + (-1.71 + 2.30i)T + (-13.4 - 45.0i)T^{2} \)
53 \( 1 + (-1.40 + 2.43i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.42 + 4.74i)T + (-49.2 + 32.4i)T^{2} \)
61 \( 1 + (-3.71 - 8.61i)T + (-41.8 + 44.3i)T^{2} \)
67 \( 1 + (4.00 - 2.01i)T + (40.0 - 53.7i)T^{2} \)
71 \( 1 + (-1.33 + 7.58i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (0.696 + 3.94i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (6.81 - 4.47i)T + (31.2 - 72.5i)T^{2} \)
83 \( 1 + (-4.30 + 2.83i)T + (32.8 - 76.2i)T^{2} \)
89 \( 1 + (0.829 + 4.70i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-7.98 - 8.45i)T + (-5.64 + 96.8i)T^{2} \)
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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

−10.41203944564544198606574250213, −9.543583527620240516166685997305, −8.584086810634648265803862776504, −7.72025467555739117890057532673, −6.13960359532213676783102937433, −5.39969800923300329164676408930, −4.65235880502354717347419114724, −3.99178397494148144373001826183, −2.06158880431061492125213592608, −1.83772004359745919892174471039, 1.97626963884506027394356732174, 3.22220748215648777350462555559, 4.37330952093350331472198154828, 5.34312489579644315701888724267, 5.96140427764854824605575768910, 6.85959748768618010337062065247, 7.62962939905696625425062390813, 8.574605349921073671740548858884, 9.874074219477110808821373318758, 10.73945137995257580389971547706

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