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Properties

Label 2-3e6-27.4-c1-0-22
Degree $2$
Conductor $729$
Sign $0.597 - 0.802i$
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  + (2.54 + 0.925i)2-s + (4.07 + 3.41i)4-s + (0.290 − 1.64i)5-s + (0.383 − 0.321i)7-s + (4.48 + 7.77i)8-s + (2.26 − 3.91i)10-s + (0.333 + 1.88i)11-s + (−2.92 + 1.06i)13-s + (1.27 − 0.463i)14-s + (2.36 + 13.4i)16-s + (1.33 − 2.30i)17-s + (−2.89 − 5.02i)19-s + (6.81 − 5.71i)20-s + (−0.900 + 5.10i)22-s + (3.55 + 2.98i)23-s + ⋯
L(s)  = 1  + (1.79 + 0.654i)2-s + (2.03 + 1.70i)4-s + (0.129 − 0.736i)5-s + (0.144 − 0.121i)7-s + (1.58 + 2.74i)8-s + (0.715 − 1.23i)10-s + (0.100 + 0.569i)11-s + (−0.811 + 0.295i)13-s + (0.340 − 0.123i)14-s + (0.592 + 3.35i)16-s + (0.323 − 0.559i)17-s + (−0.664 − 1.15i)19-s + (1.52 − 1.27i)20-s + (−0.192 + 1.08i)22-s + (0.742 + 0.622i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.597 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.597 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.597 - 0.802i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (568, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.597 - 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.93187 + 1.97466i\)
\(L(\frac12)\) \(\approx\) \(3.93187 + 1.97466i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.54 - 0.925i)T + (1.53 + 1.28i)T^{2} \)
5 \( 1 + (-0.290 + 1.64i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (-0.383 + 0.321i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (-0.333 - 1.88i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (2.92 - 1.06i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-1.33 + 2.30i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.89 + 5.02i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.55 - 2.98i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (2.45 + 0.894i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (3.53 + 2.96i)T + (5.38 + 30.5i)T^{2} \)
37 \( 1 + (-2.42 + 4.20i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (10.8 - 3.94i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (1.56 + 8.86i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (5.23 - 4.39i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 - 5.43T + 53T^{2} \)
59 \( 1 + (-0.380 + 2.15i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-5.24 + 4.39i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (11.7 - 4.26i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (1.41 - 2.45i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.96 + 8.60i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.99 + 1.81i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (2.56 + 0.933i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-5.60 - 9.71i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.19 + 6.78i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.96382475143643968196364421490, −9.580089654622645785670480668077, −8.590049067979361941763585092230, −7.34606342542838957852206514418, −7.02676783036600269800700813287, −5.80113092655410401708581768148, −4.90916674433305969857189158626, −4.52192402816144599297564083927, −3.24679946899318826354899672567, −2.04884728159201422323085346359, 1.72719610411447337497956783575, 2.88235851007004392722510645854, 3.60044504096106085899931429972, 4.74341080518417587809030553533, 5.60610664474948015264237107260, 6.42337561538195606511253332711, 7.17150592008138994836949190068, 8.490771374483719794982781674458, 10.12774313891104291887877945832, 10.41339513262699314364685554939

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