Properties

Label 2-3e6-81.52-c1-0-13
Degree $2$
Conductor $729$
Sign $0.224 + 0.974i$
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  + (−2.23 − 1.46i)2-s + (2.02 + 4.70i)4-s + (1.24 + 1.32i)5-s + (2.56 − 0.299i)7-s + (1.44 − 8.21i)8-s + (−0.841 − 4.77i)10-s + (−1.04 − 3.50i)11-s + (−0.0220 − 0.377i)13-s + (−6.15 − 3.09i)14-s + (−8.24 + 8.73i)16-s + (−2.13 − 0.778i)17-s + (−1.32 + 0.480i)19-s + (−3.68 + 8.54i)20-s + (−2.80 + 9.35i)22-s + (3.51 + 0.410i)23-s + ⋯
L(s)  = 1  + (−1.57 − 1.03i)2-s + (1.01 + 2.35i)4-s + (0.557 + 0.590i)5-s + (0.969 − 0.113i)7-s + (0.512 − 2.90i)8-s + (−0.266 − 1.50i)10-s + (−0.316 − 1.05i)11-s + (−0.00610 − 0.104i)13-s + (−1.64 − 0.826i)14-s + (−2.06 + 2.18i)16-s + (−0.518 − 0.188i)17-s + (−0.303 + 0.110i)19-s + (−0.824 + 1.91i)20-s + (−0.597 + 1.99i)22-s + (0.733 + 0.0856i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.224 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.224 + 0.974i$
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: \((2,\ 729,\ (\ :1/2),\ 0.224 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.656022 - 0.521841i\)
\(L(\frac12)\) \(\approx\) \(0.656022 - 0.521841i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 + 1.46i)T + (0.792 + 1.83i)T^{2} \)
5 \( 1 + (-1.24 - 1.32i)T + (-0.290 + 4.99i)T^{2} \)
7 \( 1 + (-2.56 + 0.299i)T + (6.81 - 1.61i)T^{2} \)
11 \( 1 + (1.04 + 3.50i)T + (-9.19 + 6.04i)T^{2} \)
13 \( 1 + (0.0220 + 0.377i)T + (-12.9 + 1.50i)T^{2} \)
17 \( 1 + (2.13 + 0.778i)T + (13.0 + 10.9i)T^{2} \)
19 \( 1 + (1.32 - 0.480i)T + (14.5 - 12.2i)T^{2} \)
23 \( 1 + (-3.51 - 0.410i)T + (22.3 + 5.30i)T^{2} \)
29 \( 1 + (-5.90 + 2.96i)T + (17.3 - 23.2i)T^{2} \)
31 \( 1 + (-5.83 + 7.83i)T + (-8.89 - 29.6i)T^{2} \)
37 \( 1 + (0.463 - 0.389i)T + (6.42 - 36.4i)T^{2} \)
41 \( 1 + (2.79 - 1.84i)T + (16.2 - 37.6i)T^{2} \)
43 \( 1 + (-2.59 - 0.615i)T + (38.4 + 19.2i)T^{2} \)
47 \( 1 + (0.0900 + 0.120i)T + (-13.4 + 45.0i)T^{2} \)
53 \( 1 + (-6.72 - 11.6i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.21 - 4.06i)T + (-49.2 - 32.4i)T^{2} \)
61 \( 1 + (-2.95 + 6.84i)T + (-41.8 - 44.3i)T^{2} \)
67 \( 1 + (-9.01 - 4.52i)T + (40.0 + 53.7i)T^{2} \)
71 \( 1 + (-0.305 - 1.73i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-0.930 + 5.27i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (3.55 + 2.33i)T + (31.2 + 72.5i)T^{2} \)
83 \( 1 + (-2.17 - 1.43i)T + (32.8 + 76.2i)T^{2} \)
89 \( 1 + (-0.553 + 3.14i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (10.5 - 11.1i)T + (-5.64 - 96.8i)T^{2} \)
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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

−10.27761253689815172703035879967, −9.514402515086500001208190202806, −8.474970507590349606640124848371, −8.111328067749300796836968845562, −7.05643336904171017531036011266, −6.03761470017680478957007028676, −4.40499708978993450654315301976, −2.98223065334690863595452429823, −2.24343465108347153476032604008, −0.861107789499476875026000939091, 1.22591356465220256488229061386, 2.18400744152771357411421996553, 4.82717201211137386431905496124, 5.26793805475996885618303910930, 6.57622828325651924773250304051, 7.15530454210656999137685235611, 8.271920758193203383413144637105, 8.655866225307120495288966487110, 9.513710934840126504450577395096, 10.28146342647129107472038278828

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