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Properties

Label 2-3e6-27.7-c1-0-22
Degree $2$
Conductor $729$
Sign $-0.0581 + 0.998i$
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  + (−1.53 + 1.28i)4-s + (−0.766 − 0.642i)7-s + (−4.69 − 1.71i)13-s + (0.694 − 3.93i)16-s + (3.5 − 6.06i)19-s + (4.69 − 1.71i)25-s + 2·28-s + (−3.06 + 2.57i)31-s + (−5.5 − 9.52i)37-s + (1.38 − 7.87i)43-s + (−1.04 − 5.90i)49-s + (9.39 − 3.42i)52-s + (−0.766 − 0.642i)61-s + (4.00 + 6.92i)64-s + (−4.69 − 1.71i)67-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)4-s + (−0.289 − 0.242i)7-s + (−1.30 − 0.474i)13-s + (0.173 − 0.984i)16-s + (0.802 − 1.39i)19-s + (0.939 − 0.342i)25-s + 0.377·28-s + (−0.550 + 0.461i)31-s + (−0.904 − 1.56i)37-s + (0.211 − 1.20i)43-s + (−0.148 − 0.844i)49-s + (1.30 − 0.474i)52-s + (−0.0980 − 0.0823i)61-s + (0.500 + 0.866i)64-s + (−0.574 − 0.208i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0581 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0581 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.0581 + 0.998i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.0581 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.478790 - 0.507488i\)
\(L(\frac12)\) \(\approx\) \(0.478790 - 0.507488i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.53 - 1.28i)T^{2} \)
5 \( 1 + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (0.766 + 0.642i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (4.69 + 1.71i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (3.06 - 2.57i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (5.5 + 9.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.38 + 7.87i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (8.16 + 46.2i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (0.766 + 0.642i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (4.69 + 1.71i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (15.9 - 5.81i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.29 - 18.7i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.04436563882172404286203390651, −9.245373993513235873386847542572, −8.595155800264143065080320370482, −7.41171798224554656905746769997, −7.02798337675798648607341013894, −5.39693270257026677572625769455, −4.75073829087518742391180276142, −3.57735746289543123429592203540, −2.60294011361708084539319257101, −0.37773775011047904446030288029, 1.50235869957726741863936480270, 3.06786296204435527771850437524, 4.35666792485979001349701728828, 5.19740217125182222965565813465, 6.04572048166024365926061055872, 7.11994924562053531194727136750, 8.119174329938685102619063864012, 9.105067976721551790449149782612, 9.762766266162477135934942651623, 10.32143897633072818669263065698

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