↑

Properties

Label 2-3e5-27.4-c1-0-5
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

Related objects

Downloads

Learn more

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: $\chi_{243} (190, \cdot )$
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 \( 1 + (-0.919 - 5.21i)T + (-91.1 + 33.1i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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