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Properties

Label 2-6e3-216.83-c1-0-27
Degree $2$
Conductor $216$
Sign $0.886 + 0.462i$
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  + (1.39 + 0.245i)2-s + (0.142 − 1.72i)3-s + (1.87 + 0.684i)4-s + (0.623 − 2.36i)6-s + (2.44 + 1.41i)8-s + (−2.95 − 0.493i)9-s + (0.389 − 0.464i)11-s + (1.44 − 3.14i)12-s + (3.06 + 2.57i)16-s + (−5.96 + 3.44i)17-s + (−3.99 − 1.41i)18-s + (1.96 − 3.39i)19-s + (0.656 − 0.550i)22-s + (2.79 − 4.02i)24-s + (−0.868 + 4.92i)25-s + ⋯
L(s)  = 1  + (0.984 + 0.173i)2-s + (0.0825 − 0.996i)3-s + (0.939 + 0.342i)4-s + (0.254 − 0.967i)6-s + (0.866 + 0.500i)8-s + (−0.986 − 0.164i)9-s + (0.117 − 0.139i)11-s + (0.418 − 0.908i)12-s + (0.766 + 0.642i)16-s + (−1.44 + 0.835i)17-s + (−0.942 − 0.333i)18-s + (0.449 − 0.779i)19-s + (0.139 − 0.117i)22-s + (0.569 − 0.821i)24-s + (−0.173 + 0.984i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 + 0.462i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.886 + 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.886 + 0.462i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ 0.886 + 0.462i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04486 - 0.501135i\)
\(L(\frac12)\) \(\approx\) \(2.04486 - 0.501135i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 - 0.245i)T \)
3 \( 1 + (-0.142 + 1.72i)T \)
good5 \( 1 + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (-5.36 + 4.49i)T^{2} \)
11 \( 1 + (-0.389 + 0.464i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (5.96 - 3.44i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.96 + 3.39i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-23.7 - 19.9i)T^{2} \)
37 \( 1 + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (12.3 - 2.17i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (-5.07 - 4.25i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (36.0 - 30.2i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (4.48 + 5.34i)T + (-10.2 + 58.1i)T^{2} \)
61 \( 1 + (-46.7 + 39.2i)T^{2} \)
67 \( 1 + (2.78 + 15.8i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-8.18 + 14.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (-13.9 - 2.46i)T + (77.9 + 28.3i)T^{2} \)
89 \( 1 + (15.9 + 9.20i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-11.6 - 9.77i)T + (16.8 + 95.5i)T^{2} \)
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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

−12.44900153274200353425274132540, −11.52413374021508106683921883428, −10.81197550487131998428501551276, −9.036037500850688999652619639760, −7.952165833455224899758190323032, −6.92640140090056991141865002957, −6.17334620611550007032661986421, −4.92104492858125928580848268773, −3.37671799860644091553649376087, −1.95903260411175575856107857651, 2.50631663912867301636519632584, 3.86399166877826678921300491126, 4.79114274482801351538071608846, 5.86689907189754581973491816615, 7.08574348952185006730020595993, 8.547847592376680823097634509314, 9.745080621511315837763792024786, 10.58734205165617713016855010941, 11.49325007817125593076051096232, 12.26691865869328275539238464536

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