Properties

Label 2-3e6-81.52-c1-0-6
Degree $2$
Conductor $729$
Sign $0.328 + 0.944i$
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  + (−1.29 − 0.848i)2-s + (0.152 + 0.354i)4-s + (0.349 + 0.370i)5-s + (−3.96 + 0.463i)7-s + (−0.432 + 2.45i)8-s + (−0.136 − 0.774i)10-s + (1.09 + 3.66i)11-s + (−0.181 − 3.10i)13-s + (5.50 + 2.76i)14-s + (3.17 − 3.36i)16-s + (−1.41 − 0.513i)17-s + (6.30 − 2.29i)19-s + (−0.0778 + 0.180i)20-s + (1.69 − 5.66i)22-s + (1.17 + 0.137i)23-s + ⋯
L(s)  = 1  + (−0.912 − 0.600i)2-s + (0.0764 + 0.177i)4-s + (0.156 + 0.165i)5-s + (−1.49 + 0.175i)7-s + (−0.153 + 0.868i)8-s + (−0.0431 − 0.244i)10-s + (0.331 + 1.10i)11-s + (−0.0502 − 0.862i)13-s + (1.47 + 0.738i)14-s + (0.793 − 0.840i)16-s + (−0.342 − 0.124i)17-s + (1.44 − 0.526i)19-s + (−0.0173 + 0.0403i)20-s + (0.361 − 1.20i)22-s + (0.245 + 0.0287i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.328 + 0.944i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.328 + 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.577240 - 0.410600i\)
\(L(\frac12)\) \(\approx\) \(0.577240 - 0.410600i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.29 + 0.848i)T + (0.792 + 1.83i)T^{2} \)
5 \( 1 + (-0.349 - 0.370i)T + (-0.290 + 4.99i)T^{2} \)
7 \( 1 + (3.96 - 0.463i)T + (6.81 - 1.61i)T^{2} \)
11 \( 1 + (-1.09 - 3.66i)T + (-9.19 + 6.04i)T^{2} \)
13 \( 1 + (0.181 + 3.10i)T + (-12.9 + 1.50i)T^{2} \)
17 \( 1 + (1.41 + 0.513i)T + (13.0 + 10.9i)T^{2} \)
19 \( 1 + (-6.30 + 2.29i)T + (14.5 - 12.2i)T^{2} \)
23 \( 1 + (-1.17 - 0.137i)T + (22.3 + 5.30i)T^{2} \)
29 \( 1 + (-6.17 + 3.10i)T + (17.3 - 23.2i)T^{2} \)
31 \( 1 + (0.0276 - 0.0371i)T + (-8.89 - 29.6i)T^{2} \)
37 \( 1 + (2.33 - 1.96i)T + (6.42 - 36.4i)T^{2} \)
41 \( 1 + (-4.96 + 3.26i)T + (16.2 - 37.6i)T^{2} \)
43 \( 1 + (-0.231 - 0.0549i)T + (38.4 + 19.2i)T^{2} \)
47 \( 1 + (2.82 + 3.80i)T + (-13.4 + 45.0i)T^{2} \)
53 \( 1 + (-6.81 - 11.7i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.400 + 1.33i)T + (-49.2 - 32.4i)T^{2} \)
61 \( 1 + (0.124 - 0.288i)T + (-41.8 - 44.3i)T^{2} \)
67 \( 1 + (-4.90 - 2.46i)T + (40.0 + 53.7i)T^{2} \)
71 \( 1 + (2.03 + 11.5i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-2.70 + 15.3i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (-11.6 - 7.69i)T + (31.2 + 72.5i)T^{2} \)
83 \( 1 + (0.587 + 0.386i)T + (32.8 + 76.2i)T^{2} \)
89 \( 1 + (-2.00 + 11.3i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-8.92 + 9.46i)T + (-5.64 - 96.8i)T^{2} \)
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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

−10.06986251273376368386458153063, −9.545160820612522117273584857005, −8.906394101880943073354264445331, −7.74592950064278293478759835626, −6.81365913345522778426581969265, −5.87839612036789466636208568986, −4.76566705437931413533895344189, −3.20284421553495849932047672267, −2.37436433902276399903524737471, −0.68034864862434686182500903644, 0.957978362628039778226041428453, 3.12117737889392918575579307217, 3.86445366444088374385612445330, 5.51442876694796083393115674679, 6.53766883216132873629842728288, 6.96247975849617207035647453236, 8.077433547483631648061023479813, 8.991226617557949414717542001583, 9.457562219372637542271092173363, 10.14878939121849073566447789385

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