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Properties

Label 2-6e3-8.3-c2-0-1
Degree $2$
Conductor $216$
Sign $-0.930 - 0.365i$
Analytic cond. $5.88557$
Root an. cond. $2.42602$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.98 − 0.248i)2-s + (3.87 + 0.986i)4-s + 8.47i·5-s + 7.86i·7-s + (−7.44 − 2.92i)8-s + (2.10 − 16.8i)10-s − 12.3·11-s − 18.5i·13-s + (1.95 − 15.6i)14-s + (14.0 + 7.64i)16-s − 2.63·17-s − 11.6·19-s + (−8.35 + 32.8i)20-s + (24.4 + 3.06i)22-s + 3.72i·23-s + ⋯
L(s)  = 1  + (−0.992 − 0.124i)2-s + (0.969 + 0.246i)4-s + 1.69i·5-s + 1.12i·7-s + (−0.930 − 0.365i)8-s + (0.210 − 1.68i)10-s − 1.12·11-s − 1.42i·13-s + (0.139 − 1.11i)14-s + (0.878 + 0.477i)16-s − 0.154·17-s − 0.611·19-s + (−0.417 + 1.64i)20-s + (1.11 + 0.139i)22-s + 0.162i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.930 - 0.365i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.930 - 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.930 - 0.365i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ -0.930 - 0.365i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0996759 + 0.527286i\)
\(L(\frac12)\) \(\approx\) \(0.0996759 + 0.527286i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.98 + 0.248i)T \)
3 \( 1 \)
good5 \( 1 - 8.47iT - 25T^{2} \)
7 \( 1 - 7.86iT - 49T^{2} \)
11 \( 1 + 12.3T + 121T^{2} \)
13 \( 1 + 18.5iT - 169T^{2} \)
17 \( 1 + 2.63T + 289T^{2} \)
19 \( 1 + 11.6T + 361T^{2} \)
23 \( 1 - 3.72iT - 529T^{2} \)
29 \( 1 + 34.7iT - 841T^{2} \)
31 \( 1 - 30.1iT - 961T^{2} \)
37 \( 1 - 56.5iT - 1.36e3T^{2} \)
41 \( 1 - 47.2T + 1.68e3T^{2} \)
43 \( 1 + 74.8T + 1.84e3T^{2} \)
47 \( 1 - 25.5iT - 2.20e3T^{2} \)
53 \( 1 - 45.8iT - 2.80e3T^{2} \)
59 \( 1 - 37.0T + 3.48e3T^{2} \)
61 \( 1 - 28.6iT - 3.72e3T^{2} \)
67 \( 1 - 2.80T + 4.48e3T^{2} \)
71 \( 1 - 1.77iT - 5.04e3T^{2} \)
73 \( 1 - 42.2T + 5.32e3T^{2} \)
79 \( 1 - 38.0iT - 6.24e3T^{2} \)
83 \( 1 + 131.T + 6.88e3T^{2} \)
89 \( 1 - 90.7T + 7.92e3T^{2} \)
97 \( 1 + 22.8T + 9.40e3T^{2} \)
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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

−12.22129952746656203659963685796, −11.22647057187349948207464537283, −10.47957586067450355691862861792, −9.869405504339386567361813409159, −8.418171034607433555590067871012, −7.70010400139753540356578803801, −6.54821146387881770066561004153, −5.63374105130175172434388916474, −3.07312418794915496370100812659, −2.46190011170494151847518984884, 0.38438160925671654003619429813, 1.85025832797878658828324058858, 4.17458629308284231421457311746, 5.35078834454482047415231429471, 6.82497418694319880558337315336, 7.85917678411887573024306905893, 8.725001145923318040500390025485, 9.523250217953448671087409263121, 10.54147135559254620401344589390, 11.49443442332653500559994991301

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