Properties

Label 2-3e5-27.7-c1-0-6
Degree $2$
Conductor $243$
Sign $0.690 + 0.723i$
Analytic cond. $1.94036$
Root an. cond. $1.39296$
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  + (2.25 − 0.821i)2-s + (2.88 − 2.42i)4-s + (0.0161 + 0.0916i)5-s + (−0.444 − 0.372i)7-s + (2.12 − 3.67i)8-s + (0.111 + 0.193i)10-s + (−0.537 + 3.04i)11-s + (−3.94 − 1.43i)13-s + (−1.30 − 0.476i)14-s + (0.462 − 2.62i)16-s + (0.995 + 1.72i)17-s + (1.92 − 3.33i)19-s + (0.268 + 0.225i)20-s + (1.28 + 7.31i)22-s + (−3.41 + 2.86i)23-s + ⋯
L(s)  = 1  + (1.59 − 0.580i)2-s + (1.44 − 1.21i)4-s + (0.00722 + 0.0409i)5-s + (−0.167 − 0.140i)7-s + (0.750 − 1.29i)8-s + (0.0353 + 0.0612i)10-s + (−0.161 + 0.918i)11-s + (−1.09 − 0.398i)13-s + (−0.349 − 0.127i)14-s + (0.115 − 0.655i)16-s + (0.241 + 0.418i)17-s + (0.441 − 0.764i)19-s + (0.0600 + 0.0504i)20-s + (0.275 + 1.55i)22-s + (−0.711 + 0.596i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(243\)    =    \(3^{5}\)
Sign: $0.690 + 0.723i$
Analytic conductor: \(1.94036\)
Root analytic conductor: \(1.39296\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{243} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 243,\ (\ :1/2),\ 0.690 + 0.723i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.46466 - 1.05552i\)
\(L(\frac12)\) \(\approx\) \(2.46466 - 1.05552i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.25 + 0.821i)T + (1.53 - 1.28i)T^{2} \)
5 \( 1 + (-0.0161 - 0.0916i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (0.444 + 0.372i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (0.537 - 3.04i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (3.94 + 1.43i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-0.995 - 1.72i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.92 + 3.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.41 - 2.86i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (6.01 - 2.18i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-1.26 + 1.06i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (2.01 + 3.49i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.03 - 0.374i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.19 - 6.79i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-2.75 - 2.30i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 - 5.40T + 53T^{2} \)
59 \( 1 + (1.78 + 10.1i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (10.1 + 8.48i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-8.30 - 3.02i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (0.572 + 0.991i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.0977 - 0.169i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.77 + 2.46i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-14.0 + 5.09i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-0.776 + 1.34i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.919 + 5.21i)T + (-91.1 - 33.1i)T^{2} \)
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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

−12.31844030070040487537094189759, −11.31586866300885617833753951132, −10.37770403605537611606240326347, −9.462321872230069210960162533680, −7.69436996042446211208626723206, −6.67627370748734418168031654354, −5.40844747385079450405653971701, −4.61946175485010708695758801366, −3.37113586732984980446872025261, −2.14205171046907675280658654757, 2.68034943383236000876671450725, 3.85740369860955283164697597208, 5.07011098078605923617418128668, 5.88221874290866488864653089741, 6.95304527341191973434782820090, 7.88800369649033163915692494050, 9.268765849621402182071578048288, 10.57147807181156775522943569168, 11.85935074558949052614321735144, 12.27330572943937035966307636009

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