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Properties

Label 2-6e3-27.25-c1-0-1
Degree $2$
Conductor $216$
Sign $-0.837 - 0.545i$
Analytic cond. $1.72476$
Root an. cond. $1.31330$
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  + (−0.198 + 1.72i)3-s + (−2.79 + 2.34i)5-s + (0.843 − 0.306i)7-s + (−2.92 − 0.683i)9-s + (−4.12 − 3.45i)11-s + (0.560 + 3.17i)13-s + (−3.48 − 5.28i)15-s + (3.01 + 5.22i)17-s + (−1.24 + 2.15i)19-s + (0.360 + 1.51i)21-s + (3.10 + 1.13i)23-s + (1.44 − 8.22i)25-s + (1.75 − 4.89i)27-s + (−1.45 + 8.27i)29-s + (6.00 + 2.18i)31-s + ⋯
L(s)  = 1  + (−0.114 + 0.993i)3-s + (−1.25 + 1.05i)5-s + (0.318 − 0.116i)7-s + (−0.973 − 0.227i)9-s + (−1.24 − 1.04i)11-s + (0.155 + 0.881i)13-s + (−0.899 − 1.36i)15-s + (0.731 + 1.26i)17-s + (−0.284 + 0.493i)19-s + (0.0787 + 0.329i)21-s + (0.647 + 0.235i)23-s + (0.289 − 1.64i)25-s + (0.337 − 0.941i)27-s + (−0.271 + 1.53i)29-s + (1.07 + 0.392i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 - 0.545i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.837 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.837 - 0.545i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ -0.837 - 0.545i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.205013 + 0.690486i\)
\(L(\frac12)\) \(\approx\) \(0.205013 + 0.690486i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.198 - 1.72i)T \)
good5 \( 1 + (2.79 - 2.34i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (-0.843 + 0.306i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (4.12 + 3.45i)T + (1.91 + 10.8i)T^{2} \)
13 \( 1 + (-0.560 - 3.17i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-3.01 - 5.22i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.24 - 2.15i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.10 - 1.13i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (1.45 - 8.27i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-6.00 - 2.18i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (-0.854 - 1.47i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.766 + 4.34i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (0.679 + 0.569i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (2.22 - 0.809i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 - 3.93T + 53T^{2} \)
59 \( 1 + (-9.65 + 8.10i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-5.55 + 2.02i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.25 + 7.11i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (0.922 + 1.59i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.38 + 2.39i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.942 - 5.34i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (1.46 - 8.33i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (6.85 - 11.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.16 + 6.84i)T + (16.8 + 95.5i)T^{2} \)
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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

−12.47337030676671045316479910057, −11.32681243876683670601579063869, −10.89153498013190651237853622155, −10.15952587131436902994851965290, −8.573088172925194939641025823462, −7.939929318912523616813357394626, −6.58748262068979650711493605294, −5.26965141741751037558879772572, −3.90391520903319099776679127086, −3.11249980884065300200037895228, 0.61807186295700045001756433094, 2.70030297359246340495916209382, 4.61646024897919094939606513862, 5.45756824579274569084823642719, 7.21848760638084630449713564598, 7.87124533431905207313189415399, 8.523733675493557526111706180184, 9.979452035291464471275364032438, 11.42680867746565840268635283451, 11.93745338900833730664686933087

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