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Properties

Label 2-3e5-9.5-c2-0-9
Degree $2$
Conductor $243$
Sign $0.766 - 0.642i$
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  + (1.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s + (6 + 3.46i)5-s + (0.5 + 0.866i)7-s + 8.66i·8-s + 12·10-s + (−12 + 6.92i)11-s + (−5.5 + 9.52i)13-s + (1.5 + 0.866i)14-s + (5.5 + 9.52i)16-s − 20.7i·17-s + 35·19-s + (−5.99 + 3.46i)20-s + (−12 + 20.7i)22-s + (6 + 3.46i)23-s + ⋯
L(s)  = 1  + (0.750 − 0.433i)2-s + (−0.125 + 0.216i)4-s + (1.20 + 0.692i)5-s + (0.0714 + 0.123i)7-s + 1.08i·8-s + 1.20·10-s + (−1.09 + 0.629i)11-s + (−0.423 + 0.732i)13-s + (0.107 + 0.0618i)14-s + (0.343 + 0.595i)16-s − 1.22i·17-s + 1.84·19-s + (−0.299 + 0.173i)20-s + (−0.545 + 0.944i)22-s + (0.260 + 0.150i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(243\)    =    \(3^{5}\)
Sign: $0.766 - 0.642i$
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: \((2,\ 243,\ (\ :1),\ 0.766 - 0.642i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.31443 + 0.842384i\)
\(L(\frac12)\) \(\approx\) \(2.31443 + 0.842384i\)
\(L(2)\) 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 \( 1 + (-6 - 3.46i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (12 - 6.92i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (5.5 - 9.52i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 20.7iT - 289T^{2} \)
19 \( 1 - 35T + 361T^{2} \)
23 \( 1 + (-6 - 3.46i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-24 + 13.8i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-12.5 + 21.6i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 35T + 1.36e3T^{2} \)
41 \( 1 + (66 + 38.1i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-18.5 - 32.0i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (48 - 27.7i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 41.5iT - 2.80e3T^{2} \)
59 \( 1 + (-6 - 3.46i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (37 + 64.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-11 + 19.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 83.1iT - 5.04e3T^{2} \)
73 \( 1 - 2T + 5.32e3T^{2} \)
79 \( 1 + (-36.5 - 63.2i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-6 + 3.46i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 62.3iT - 7.92e3T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.97850395148594652092409009023, −11.30312955681253992571963281928, −9.993345259607126744209131517834, −9.465923812956025430893007962220, −7.921929144781305130377577997000, −6.90288776115135209612658406446, −5.48742530793269106489739808755, −4.77942012316488063751754149956, −3.02850370797812425897872966736, −2.24106474784551606867477875669, 1.14595377085938302624036586244, 3.12062856852199609452743453960, 4.89679611519374503348306914215, 5.44355733459511986709593205456, 6.29035315922526916260661081823, 7.72940124129270215043332669705, 8.910510287622884519845330055148, 10.00206032502146487627040238552, 10.49542155419057272550633242385, 12.16720256043048746622969001990

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