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Properties

Label 2-3e5-9.4-c1-0-8
Degree $2$
Conductor $243$
Sign $-0.173 + 0.984i$
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  + (−0.439 − 0.761i)2-s + (0.613 − 1.06i)4-s + (1.93 − 3.35i)5-s + (1.09 + 1.89i)7-s − 2.83·8-s − 3.41·10-s + (0.0812 + 0.140i)11-s + (−1.20 + 2.08i)13-s + (0.960 − 1.66i)14-s + (0.0209 + 0.0362i)16-s − 3·17-s + 3.59·19-s + (−2.37 − 4.12i)20-s + (0.0714 − 0.123i)22-s + (1.41 − 2.45i)23-s + ⋯
L(s)  = 1  + (−0.310 − 0.538i)2-s + (0.306 − 0.531i)4-s + (0.867 − 1.50i)5-s + (0.412 + 0.715i)7-s − 1.00·8-s − 1.07·10-s + (0.0244 + 0.0424i)11-s + (−0.334 + 0.579i)13-s + (0.256 − 0.444i)14-s + (0.00523 + 0.00906i)16-s − 0.727·17-s + 0.825·19-s + (−0.532 − 0.921i)20-s + (0.0152 − 0.0263i)22-s + (0.295 − 0.512i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(243\)    =    \(3^{5}\)
Sign: $-0.173 + 0.984i$
Analytic conductor: \(1.94036\)
Root analytic conductor: \(1.39296\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{243} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 243,\ (\ :1/2),\ -0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.821347 - 0.978843i\)
\(L(\frac12)\) \(\approx\) \(0.821347 - 0.978843i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.439 + 0.761i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.93 + 3.35i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.09 - 1.89i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.0812 - 0.140i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.20 - 2.08i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 - 3.59T + 19T^{2} \)
23 \( 1 + (-1.41 + 2.45i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.35 - 5.81i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.59 + 4.49i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.63T + 37T^{2} \)
41 \( 1 + (2.90 - 5.02i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.11 - 5.39i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.69 - 6.40i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 1.40T + 53T^{2} \)
59 \( 1 + (2.56 - 4.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.89 - 3.27i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.93 + 5.07i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 15.3T + 71T^{2} \)
73 \( 1 - 8.68T + 73T^{2} \)
79 \( 1 + (-0.634 - 1.09i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.23 + 7.34i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 7.72T + 89T^{2} \)
97 \( 1 + (-1.95 - 3.38i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.93390346047083319484876391885, −10.89102861349061897008895545373, −9.689239237666589257472029452916, −9.175485335088590598372321744976, −8.374452726107497352684240023566, −6.59099089089800989513481856479, −5.49869552792536264479224957298, −4.74564252061874689332803349839, −2.43165677394746859561279716584, −1.30247530293544449143022274277, 2.42459857772813078991917157182, 3.55766295003154750395776900164, 5.49214105282758940719463297645, 6.72494807826469105845938039041, 7.17650957452220521544654321273, 8.224274996074344934032420626365, 9.552836615781079910720782443445, 10.48423272686725060546142238474, 11.20789941860871652336842277559, 12.27701011409023295769953813227

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