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Properties

Label 2-3e6-27.7-c1-0-19
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  + (−0.730 + 0.266i)2-s + (−1.06 + 0.896i)4-s + (0.412 + 2.34i)5-s + (−1.91 − 1.60i)7-s + (1.32 − 2.28i)8-s + (−0.924 − 1.60i)10-s + (0.545 − 3.09i)11-s + (−1.25 − 0.457i)13-s + (1.82 + 0.665i)14-s + (0.127 − 0.725i)16-s + (−3.13 − 5.43i)17-s + (−4.03 + 6.98i)19-s + (−2.53 − 2.13i)20-s + (0.424 + 2.40i)22-s + (3.10 − 2.60i)23-s + ⋯
L(s)  = 1  + (−0.516 + 0.188i)2-s + (−0.534 + 0.448i)4-s + (0.184 + 1.04i)5-s + (−0.724 − 0.607i)7-s + (0.466 − 0.808i)8-s + (−0.292 − 0.506i)10-s + (0.164 − 0.932i)11-s + (−0.348 − 0.126i)13-s + (0.488 + 0.177i)14-s + (0.0319 − 0.181i)16-s + (−0.760 − 1.31i)17-s + (−0.925 + 1.60i)19-s + (−0.567 − 0.476i)20-s + (0.0904 + 0.512i)22-s + (0.647 − 0.543i)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: $\chi_{729} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.597 + 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.560059 - 0.281272i\)
\(L(\frac12)\) \(\approx\) \(0.560059 - 0.281272i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.730 - 0.266i)T + (1.53 - 1.28i)T^{2} \)
5 \( 1 + (-0.412 - 2.34i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (1.91 + 1.60i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (-0.545 + 3.09i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (1.25 + 0.457i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (3.13 + 5.43i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.03 - 6.98i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.10 + 2.60i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-8.72 + 3.17i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-2.16 + 1.82i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (2.76 + 4.79i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.67 - 2.43i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.405 + 2.30i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (3.53 + 2.96i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 - 0.135T + 53T^{2} \)
59 \( 1 + (0.694 + 3.93i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-0.261 - 0.219i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (9.51 + 3.46i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (4.09 + 7.09i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.15 + 10.6i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.83 - 1.39i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (0.858 - 0.312i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-1.86 + 3.22i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.04 - 5.90i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.27066065393593114606202895077, −9.429250684385576779044202899661, −8.559256204241471492899157091446, −7.67779015967810796444228798762, −6.78879269436789340719499250193, −6.22843475266527430860714384367, −4.63714069067004768812527836282, −3.59829758551640076492950975899, −2.71549873705370943526757128435, −0.42964314180149260454168666150, 1.28332672869212795329662540436, 2.55728499087470457597795258589, 4.45825288084052972659872001939, 4.87861225326813497645859850432, 6.06901079421703887708844124477, 6.99021163418058984152807193647, 8.501811454664952951556822631063, 8.832836492200927983154804406435, 9.517242481562749413873802645797, 10.29561340268701925537771210453

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