Properties

Label 2-3e6-9.4-c1-0-27
Degree $2$
Conductor $729$
Sign $-1$
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  + (−0.527 − 0.913i)2-s + (0.444 − 0.769i)4-s + (0.872 − 1.51i)5-s + (−1.22 − 2.12i)7-s − 3.04·8-s − 1.84·10-s + (0.627 + 1.08i)11-s + (2.27 − 3.93i)13-s + (−1.29 + 2.24i)14-s + (0.716 + 1.24i)16-s − 6.64·17-s + 0.249·19-s + (−0.775 − 1.34i)20-s + (0.661 − 1.14i)22-s + (−0.421 + 0.729i)23-s + ⋯
L(s)  = 1  + (−0.372 − 0.645i)2-s + (0.222 − 0.384i)4-s + (0.390 − 0.676i)5-s + (−0.464 − 0.804i)7-s − 1.07·8-s − 0.582·10-s + (0.189 + 0.327i)11-s + (0.630 − 1.09i)13-s + (−0.346 + 0.600i)14-s + (0.179 + 0.310i)16-s − 1.61·17-s + 0.0571·19-s + (−0.173 − 0.300i)20-s + (0.140 − 0.244i)22-s + (−0.0877 + 0.152i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-1$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03219i\)
\(L(\frac12)\) \(\approx\) \(1.03219i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.527 + 0.913i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.872 + 1.51i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.22 + 2.12i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.627 - 1.08i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.27 + 3.93i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 6.64T + 17T^{2} \)
19 \( 1 - 0.249T + 19T^{2} \)
23 \( 1 + (0.421 - 0.729i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.256 - 0.443i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.410 + 0.710i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.60T + 37T^{2} \)
41 \( 1 + (-4.07 + 7.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.16 + 3.74i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.65 - 4.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 + (-1.50 + 2.60i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.44 + 2.49i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.04 - 8.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.0894T + 71T^{2} \)
73 \( 1 + 5.32T + 73T^{2} \)
79 \( 1 + (2.38 + 4.13i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.02 + 6.96i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6.70T + 89T^{2} \)
97 \( 1 + (2.74 + 4.75i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.05635863512140669046883812691, −9.239222456721195254313498311829, −8.611624232022514186331161618235, −7.31626692163085868640353603372, −6.37277920403888121652973632023, −5.53369000776621485392560280703, −4.34377026256273973272644082075, −3.11281594885382635757352578998, −1.79643752401356964227417481285, −0.57693279686071007175702932035, 2.22192789381039815608951698888, 3.12456900298000725644776433212, 4.43309039349044947179248004439, 6.15042213490454612861795348314, 6.31114851557478658734128954283, 7.16988959514259210232032775274, 8.379810723645269485083599518101, 8.967834974027086461177210431995, 9.650337126914099419336180987509, 10.95450235084996937674413289009

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