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Properties

Label 2-3e5-9.4-c1-0-2
Degree $2$
Conductor $243$
Sign $-0.939 - 0.342i$
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  + (1.22 + 2.12i)2-s + (−1.99 + 3.46i)4-s + (−1.22 + 2.12i)5-s + (−1 − 1.73i)7-s − 4.89·8-s − 5.99·10-s + (1.22 + 2.12i)11-s + (0.5 − 0.866i)13-s + (2.44 − 4.24i)14-s + (−1.99 − 3.46i)16-s + 7.34·17-s − 19-s + (−4.89 − 8.48i)20-s + (−2.99 + 5.19i)22-s + (−1.22 + 2.12i)23-s + ⋯
L(s)  = 1  + (0.866 + 1.49i)2-s + (−0.999 + 1.73i)4-s + (−0.547 + 0.948i)5-s + (−0.377 − 0.654i)7-s − 1.73·8-s − 1.89·10-s + (0.369 + 0.639i)11-s + (0.138 − 0.240i)13-s + (0.654 − 1.13i)14-s + (−0.499 − 0.866i)16-s + 1.78·17-s − 0.229·19-s + (−1.09 − 1.89i)20-s + (−0.639 + 1.10i)22-s + (−0.255 + 0.442i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(243\)    =    \(3^{5}\)
Sign: $-0.939 - 0.342i$
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: \((2,\ 243,\ (\ :1/2),\ -0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.281850 + 1.59845i\)
\(L(\frac12)\) \(\approx\) \(0.281850 + 1.59845i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.22 - 2.12i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.22 - 2.12i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.22 - 2.12i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 7.34T + 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 + (1.22 - 2.12i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.44 + 4.24i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + (-2.44 + 4.24i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.5 + 9.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.89 - 8.48i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 7.34T + 53T^{2} \)
59 \( 1 + (1.22 - 2.12i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.34T + 71T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 + (-3.5 - 6.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.12 + 10.6i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (-3.5 - 6.06i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.78382718779806332095657786738, −11.90156826864929641633952573201, −10.62127709542032432156196099098, −9.538740833094963594217257032437, −7.905980926983958543144764963989, −7.44785567565744640055619012009, −6.56507115457489684303867631806, −5.57891235047954174208119549601, −4.15757646356392193107558464887, −3.35378781494504489688477699309, 1.16014551713560450284103046941, 2.93561612945166242457854500021, 3.98319249030340392404308307381, 5.08236173248571196985054468629, 6.06576467313299219411260582110, 8.037502469034969224991701045018, 9.098012415525434396990696620925, 9.934710365801532716528600136060, 11.09267955930924400432353718500, 11.92137185682653318876118917375

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