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Properties

Label 2-3e5-9.5-c2-0-17
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 + 3.46i)4-s + (−5.5 − 9.52i)7-s + (11 − 19.0i)13-s + (−7.99 − 13.8i)16-s + 26·19-s + (−12.5 − 21.6i)25-s + 44·28-s + (6.5 − 11.2i)31-s − 73·37-s + (−41.5 − 71.8i)43-s + (−36 + 62.3i)49-s + (44 + 76.2i)52-s + (60.5 + 104. i)61-s + 63.9·64-s + (54.5 − 94.3i)67-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)4-s + (−0.785 − 1.36i)7-s + (0.846 − 1.46i)13-s + (−0.499 − 0.866i)16-s + 1.36·19-s + (−0.5 − 0.866i)25-s + 1.57·28-s + (0.209 − 0.363i)31-s − 1.97·37-s + (−0.965 − 1.67i)43-s + (−0.734 + 1.27i)49-s + (0.846 + 1.46i)52-s + (0.991 + 1.71i)61-s + 0.999·64-s + (0.813 − 1.40i)67-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: $\chi_{243} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 243,\ (\ :1),\ 0.173 + 0.984i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.768932 - 0.645210i\)
\(L(\frac12)\) \(\approx\) \(0.768932 - 0.645210i\)
\(L(2)\) 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 \( 1 + (2 - 3.46i)T^{2} \)
5 \( 1 + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (5.5 + 9.52i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-11 + 19.0i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 289T^{2} \)
19 \( 1 - 26T + 361T^{2} \)
23 \( 1 + (264.5 + 458. i)T^{2} \)
29 \( 1 + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-6.5 + 11.2i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 73T + 1.36e3T^{2} \)
41 \( 1 + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (41.5 + 71.8i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 2.80e3T^{2} \)
59 \( 1 + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-60.5 - 104. i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-54.5 + 94.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 97T + 5.32e3T^{2} \)
79 \( 1 + (-71 - 122. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.81665279896007131272708491773, −10.52023409708469375586464216427, −9.886575073449185070821132025944, −8.613350997680044264048085333605, −7.71208101889481213688889481271, −6.86082296424511453787695535562, −5.38328873934200320089991809104, −3.89206492771762718266429759829, −3.22215371327991025871540503024, −0.55682468455908070927384990622, 1.69634415217055647026360914582, 3.44616630578991031180337498466, 5.01109181839463085790046077550, 5.92536824276041341363662873232, 6.81021228481683706497037587427, 8.560869071073504338394998332409, 9.274289860066989745310240649863, 9.883017161406670449205910830109, 11.26435442618341913003505498507, 11.97443947961758062277571903882

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