↑

Properties

Label 2-3e5-9.2-c2-0-22
Degree $2$
Conductor $243$
Sign $-0.173 + 0.984i$
Analytic cond. $6.62127$
Root an. cond. $2.57318$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + (−7.5 + 4.33i)5-s + (5 − 8.66i)7-s − 8.66i·8-s − 15·10-s + (−12 − 6.92i)11-s + (−1 − 1.73i)13-s + (15 − 8.66i)14-s + (5.5 − 9.52i)16-s − 10.3i·17-s − 19-s + (7.49 + 4.33i)20-s + (−12 − 20.7i)22-s + (−7.5 + 4.33i)23-s + ⋯
L(s)  = 1  + (0.750 + 0.433i)2-s + (−0.125 − 0.216i)4-s + (−1.5 + 0.866i)5-s + (0.714 − 1.23i)7-s − 1.08i·8-s − 1.5·10-s + (−1.09 − 0.629i)11-s + (−0.0769 − 0.133i)13-s + (1.07 − 0.618i)14-s + (0.343 − 0.595i)16-s − 0.611i·17-s − 0.0526·19-s + (0.375 + 0.216i)20-s + (−0.545 − 0.944i)22-s + (−0.326 + 0.188i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(243\)    =    \(3^{5}\)
Sign: $-0.173 + 0.984i$
Analytic conductor: \(6.62127\)
Root analytic conductor: \(2.57318\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{243} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 243,\ (\ :1),\ -0.173 + 0.984i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.678126 - 0.808159i\)
\(L(\frac12)\) \(\approx\) \(0.678126 - 0.808159i\)
\(L(2)\) 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 \( 1 + (-1.5 - 0.866i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (7.5 - 4.33i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (-5 + 8.66i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (12 + 6.92i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 10.3iT - 289T^{2} \)
19 \( 1 + T + 361T^{2} \)
23 \( 1 + (7.5 - 4.33i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-10.5 - 6.06i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (10 + 17.3i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 28T + 1.36e3T^{2} \)
41 \( 1 + (12 - 6.92i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (31 - 53.6i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-46.5 - 26.8i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 67.5iT - 2.80e3T^{2} \)
59 \( 1 + (-33 + 19.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-17 + 29.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (2.5 + 4.33i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 5.19iT - 5.04e3T^{2} \)
73 \( 1 - 137T + 5.32e3T^{2} \)
79 \( 1 + (40 - 69.2i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-60 - 34.6i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 155. iT - 7.92e3T^{2} \)
97 \( 1 + (8.5 - 14.7i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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

−11.46352640274224493205074757454, −10.84329611286192506556137410835, −10.01711684930875242974015312672, −8.189266986890722782145298478797, −7.50614466264138464406037884242, −6.69037024060955413525342318979, −5.17662276271383374752745810065, −4.20923863597470108327135352683, −3.27539057862633224302909101759, −0.43607992286639274612823851598, 2.24667305657941093531123060353, 3.73407826202446150033603096743, 4.77993895542070262915307961210, 5.39428666028600817467827576703, 7.49790695710384172454927139731, 8.354974032637061007982361996039, 8.801826681632423345683555830146, 10.61754055810147484548078652672, 11.67772434463158152363836680091, 12.25010844620087369405750146056

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