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Properties

Label 2-3e6-27.25-c1-0-22
Degree $2$
Conductor $729$
Sign $0.973 + 0.230i$
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  + (1.87 + 0.684i)4-s + (3.75 − 1.36i)7-s + (−1.21 − 6.89i)13-s + (3.06 + 2.57i)16-s + (0.5 − 0.866i)19-s + (−0.868 + 4.92i)25-s + 8·28-s + (−10.3 − 3.76i)31-s + (5 + 8.66i)37-s + (3.83 + 3.21i)43-s + (6.89 − 5.78i)49-s + (2.43 − 13.7i)52-s + (0.939 − 0.342i)61-s + (4.00 + 6.92i)64-s + (0.868 + 4.92i)67-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)4-s + (1.42 − 0.517i)7-s + (−0.337 − 1.91i)13-s + (0.766 + 0.642i)16-s + (0.114 − 0.198i)19-s + (−0.173 + 0.984i)25-s + 1.51·28-s + (−1.85 − 0.675i)31-s + (0.821 + 1.42i)37-s + (0.584 + 0.490i)43-s + (0.984 − 0.826i)49-s + (0.337 − 1.91i)52-s + (0.120 − 0.0437i)61-s + (0.500 + 0.866i)64-s + (0.106 + 0.601i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.973 + 0.230i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.973 + 0.230i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.13599 - 0.249661i\)
\(L(\frac12)\) \(\approx\) \(2.13599 - 0.249661i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.87 - 0.684i)T^{2} \)
5 \( 1 + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (-3.75 + 1.36i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (1.91 + 10.8i)T^{2} \)
13 \( 1 + (1.21 + 6.89i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (10.3 + 3.76i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (-5 - 8.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-3.83 - 3.21i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (36.0 - 30.2i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-0.939 + 0.342i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-0.868 - 4.92i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.25 - 12.8i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.83 - 3.21i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.70925226193231354456272021353, −9.634733782855660582524628932463, −8.251524337115431433441680777913, −7.77599912709894921442483145417, −7.14853762790095540475021403753, −5.81323187063364578738272510163, −5.04902550145165744703040859174, −3.73357354768251862478401609262, −2.61517623242230174196041248963, −1.30855079418430445403231739178, 1.67746716554469764605378563990, 2.31871220956846850570265009291, 4.05019558106758557138363098820, 5.08023397356834089827238608801, 5.97779230704522323554785529621, 7.00043667889031147793486446017, 7.69012995415580820464636569990, 8.754391096303050862800700849369, 9.509916607610505459919290015648, 10.69916940274756016267182009297

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