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Properties

Label 2-6e3-8.3-c2-0-21
Degree $2$
Conductor $216$
Sign $0.948 + 0.317i$
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  + (0.808 + 1.82i)2-s + (−2.69 + 2.95i)4-s − 5.51i·5-s − 11.2i·7-s + (−7.58 − 2.53i)8-s + (10.0 − 4.45i)10-s + 17.9·11-s − 5.65i·13-s + (20.5 − 9.06i)14-s + (−1.49 − 15.9i)16-s + 9.53·17-s − 33.1·19-s + (16.3 + 14.8i)20-s + (14.4 + 32.7i)22-s + 19.3i·23-s + ⋯
L(s)  = 1  + (0.404 + 0.914i)2-s + (−0.673 + 0.739i)4-s − 1.10i·5-s − 1.60i·7-s + (−0.948 − 0.317i)8-s + (1.00 − 0.445i)10-s + 1.62·11-s − 0.435i·13-s + (1.46 − 0.647i)14-s + (−0.0932 − 0.995i)16-s + 0.561·17-s − 1.74·19-s + (0.815 + 0.742i)20-s + (0.658 + 1.49i)22-s + 0.843i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.317i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.948 + 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.948 + 0.317i$
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: \((2,\ 216,\ (\ :1),\ 0.948 + 0.317i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.66085 - 0.270275i\)
\(L(\frac12)\) \(\approx\) \(1.66085 - 0.270275i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.808 - 1.82i)T \)
3 \( 1 \)
good5 \( 1 + 5.51iT - 25T^{2} \)
7 \( 1 + 11.2iT - 49T^{2} \)
11 \( 1 - 17.9T + 121T^{2} \)
13 \( 1 + 5.65iT - 169T^{2} \)
17 \( 1 - 9.53T + 289T^{2} \)
19 \( 1 + 33.1T + 361T^{2} \)
23 \( 1 - 19.3iT - 529T^{2} \)
29 \( 1 + 29.8iT - 841T^{2} \)
31 \( 1 - 6.52iT - 961T^{2} \)
37 \( 1 + 33.9iT - 1.36e3T^{2} \)
41 \( 1 - 56.8T + 1.68e3T^{2} \)
43 \( 1 - 19.1T + 1.84e3T^{2} \)
47 \( 1 - 30.9iT - 2.20e3T^{2} \)
53 \( 1 + 11.2iT - 2.80e3T^{2} \)
59 \( 1 + 10.2T + 3.48e3T^{2} \)
61 \( 1 + 3.18iT - 3.72e3T^{2} \)
67 \( 1 - 13.1T + 4.48e3T^{2} \)
71 \( 1 - 103. iT - 5.04e3T^{2} \)
73 \( 1 + 21.3T + 5.32e3T^{2} \)
79 \( 1 - 134. iT - 6.24e3T^{2} \)
83 \( 1 + 56.2T + 6.88e3T^{2} \)
89 \( 1 + 114.T + 7.92e3T^{2} \)
97 \( 1 - 126.T + 9.40e3T^{2} \)
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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

−12.46211005794019774624130881605, −11.20526666926608940274406840892, −9.815293699859759715464658076198, −8.892500499029978390570552408445, −7.910628931325957394857993829592, −6.93318190105135639345330221147, −5.86499607520593794899482901722, −4.39157307290862973329957813617, −3.91052294372761114597556450348, −0.893535173004069934274387082972, 1.97497591561817667279948860852, 3.09063747495206542479741044167, 4.40965700922481392074494106429, 5.97415561447457549529803254283, 6.61710378550003088879925345614, 8.637560596224185278919098554389, 9.243963570686004832290051249629, 10.43442195137036395824037981561, 11.31424645316156823632865307041, 12.07937115728454139710364835790

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