Properties

Label 2-3e5-27.25-c1-0-3
Degree $2$
Conductor $243$
Sign $0.108 - 0.994i$
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  + (−0.291 + 1.65i)2-s + (−0.775 − 0.282i)4-s + (0.865 − 0.726i)5-s + (3.67 − 1.33i)7-s + (−0.987 + 1.70i)8-s + (0.949 + 1.64i)10-s + (1.43 + 1.20i)11-s + (−0.127 − 0.721i)13-s + (1.14 + 6.47i)14-s + (−3.80 − 3.19i)16-s + (0.944 + 1.63i)17-s + (−1.37 + 2.37i)19-s + (−0.876 + 0.318i)20-s + (−2.40 + 2.02i)22-s + (−5.47 − 1.99i)23-s + ⋯
L(s)  = 1  + (−0.206 + 1.17i)2-s + (−0.387 − 0.141i)4-s + (0.386 − 0.324i)5-s + (1.38 − 0.505i)7-s + (−0.349 + 0.604i)8-s + (0.300 + 0.519i)10-s + (0.432 + 0.362i)11-s + (−0.0352 − 0.200i)13-s + (0.304 + 1.72i)14-s + (−0.951 − 0.798i)16-s + (0.229 + 0.396i)17-s + (−0.314 + 0.544i)19-s + (−0.195 + 0.0713i)20-s + (−0.513 + 0.430i)22-s + (−1.14 − 0.415i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.108 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.108 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(243\)    =    \(3^{5}\)
Sign: $0.108 - 0.994i$
Analytic conductor: \(1.94036\)
Root analytic conductor: \(1.39296\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{243} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 243,\ (\ :1/2),\ 0.108 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02047 + 0.915280i\)
\(L(\frac12)\) \(\approx\) \(1.02047 + 0.915280i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.291 - 1.65i)T + (-1.87 - 0.684i)T^{2} \)
5 \( 1 + (-0.865 + 0.726i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (-3.67 + 1.33i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (-1.43 - 1.20i)T + (1.91 + 10.8i)T^{2} \)
13 \( 1 + (0.127 + 0.721i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-0.944 - 1.63i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.37 - 2.37i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.47 + 1.99i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.923 + 5.23i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (1.25 + 0.458i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (1.69 + 2.94i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.311 + 1.76i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-3.85 - 3.23i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-1.60 + 0.584i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + 2.84T + 53T^{2} \)
59 \( 1 + (-8.62 + 7.23i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (4.91 - 1.78i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.328 + 1.86i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (6.09 + 10.5i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.94 - 8.56i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.14 + 12.1i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-2.02 + 11.5i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (2.86 - 4.96i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.263 + 0.220i)T + (16.8 + 95.5i)T^{2} \)
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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

−12.27629623074716349235095691394, −11.39571912855377269940655418569, −10.32919407759941719240123593508, −9.112443057752681637685263600712, −8.091251757998967926237122727581, −7.56901882496356080724770939854, −6.29647398083871434903959439988, −5.35617424433274890772424818148, −4.21949133809695564892264385727, −1.88115888564932648029061331053, 1.56578991423879610113194254969, 2.68094339319016797955751080011, 4.20928260103635519134580507024, 5.60813153896115777344172907641, 6.85414021135578182817751478737, 8.271215129765973911939729326766, 9.177473081653695389096116597515, 10.20368064045576707130445463314, 11.05191367896631462576221905959, 11.71508866055076222876533489299

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