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Properties

Label 2-3e6-27.13-c1-0-13
Degree $2$
Conductor $729$
Sign $-0.835 + 0.549i$
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  + (−0.439 − 2.49i)2-s + (−4.14 + 1.50i)4-s + (−0.358 − 0.300i)5-s + (3.03 + 1.10i)7-s + (3.05 + 5.28i)8-s + (−0.592 + 1.02i)10-s + (2.37 − 1.99i)11-s + (−0.379 + 2.15i)13-s + (1.41 − 8.04i)14-s + (5.08 − 4.26i)16-s + (1.5 − 2.59i)17-s + (−0.0209 − 0.0362i)19-s + (1.93 + 0.705i)20-s + (−6.02 − 5.05i)22-s + (5.73 − 2.08i)23-s + ⋯
L(s)  = 1  + (−0.310 − 1.76i)2-s + (−2.07 + 0.754i)4-s + (−0.160 − 0.134i)5-s + (1.14 + 0.417i)7-s + (1.07 + 1.86i)8-s + (−0.187 + 0.324i)10-s + (0.717 − 0.601i)11-s + (−0.105 + 0.596i)13-s + (0.379 − 2.15i)14-s + (1.27 − 1.06i)16-s + (0.363 − 0.630i)17-s + (−0.00480 − 0.00832i)19-s + (0.433 + 0.157i)20-s + (−1.28 − 1.07i)22-s + (1.19 − 0.435i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.835 + 0.549i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.835 + 0.549i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.366011 - 1.22256i\)
\(L(\frac12)\) \(\approx\) \(0.366011 - 1.22256i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.439 + 2.49i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (0.358 + 0.300i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (-3.03 - 1.10i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (-2.37 + 1.99i)T + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (0.379 - 2.15i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0209 + 0.0362i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.73 + 2.08i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.14 + 6.47i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-5.85 + 2.12i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (1.79 - 3.11i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.33 - 7.58i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (0.450 - 0.378i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (9.07 + 3.30i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + 4.95T + 53T^{2} \)
59 \( 1 + (-6.53 - 5.48i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-1.19 - 0.433i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-1.73 + 9.85i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-5.91 + 10.2i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.11 - 7.13i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.91 - 10.8i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-0.262 - 1.48i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (7.93 + 13.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-14.2 + 11.9i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.09967957898776803023951041496, −9.403120570028869218356748571413, −8.536455623366138002564135392945, −8.031434216769767004403874556081, −6.47860926988457701863563141083, −4.97162961189259660385637688532, −4.34748643567407616672423956551, −3.14360952021820085942727265727, −2.07359958268742720250439342234, −0.911335230770448958832362849511, 1.32792992481616282169851449775, 3.68136969772610726992969723728, 4.85536853846760588303286308595, 5.38271481210261889698086677718, 6.62874843009933123106824394541, 7.27810161903010871777505816521, 7.962419012127302454940843398799, 8.713208173875048968228613029030, 9.544162304236430462267243599961, 10.53529802184576089599559980535

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