Properties

Label 2-6e3-8.3-c2-0-11
Degree $2$
Conductor $216$
Sign $0.459 - 0.888i$
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  + (−0.316 + 1.97i)2-s + (−3.79 − 1.25i)4-s − 4.41i·5-s + 3.59i·7-s + (3.67 − 7.10i)8-s + (8.71 + 1.39i)10-s + 18.7·11-s + 19.9i·13-s + (−7.10 − 1.14i)14-s + (12.8 + 9.51i)16-s + 8.00·17-s + 16.1·19-s + (−5.52 + 16.7i)20-s + (−5.95 + 37.0i)22-s − 18.3i·23-s + ⋯
L(s)  = 1  + (−0.158 + 0.987i)2-s + (−0.949 − 0.312i)4-s − 0.882i·5-s + 0.514i·7-s + (0.459 − 0.888i)8-s + (0.871 + 0.139i)10-s + 1.70·11-s + 1.53i·13-s + (−0.507 − 0.0814i)14-s + (0.804 + 0.594i)16-s + 0.470·17-s + 0.849·19-s + (−0.276 + 0.838i)20-s + (−0.270 + 1.68i)22-s − 0.799i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.459 - 0.888i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.459 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.459 - 0.888i$
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: \((2,\ 216,\ (\ :1),\ 0.459 - 0.888i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.20794 + 0.735134i\)
\(L(\frac12)\) \(\approx\) \(1.20794 + 0.735134i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.316 - 1.97i)T \)
3 \( 1 \)
good5 \( 1 + 4.41iT - 25T^{2} \)
7 \( 1 - 3.59iT - 49T^{2} \)
11 \( 1 - 18.7T + 121T^{2} \)
13 \( 1 - 19.9iT - 169T^{2} \)
17 \( 1 - 8.00T + 289T^{2} \)
19 \( 1 - 16.1T + 361T^{2} \)
23 \( 1 + 18.3iT - 529T^{2} \)
29 \( 1 - 17.9iT - 841T^{2} \)
31 \( 1 + 29.7iT - 961T^{2} \)
37 \( 1 - 26.7iT - 1.36e3T^{2} \)
41 \( 1 - 40.2T + 1.68e3T^{2} \)
43 \( 1 + 71.8T + 1.84e3T^{2} \)
47 \( 1 - 23.3iT - 2.20e3T^{2} \)
53 \( 1 + 90.7iT - 2.80e3T^{2} \)
59 \( 1 - 10.9T + 3.48e3T^{2} \)
61 \( 1 - 90.9iT - 3.72e3T^{2} \)
67 \( 1 - 74.0T + 4.48e3T^{2} \)
71 \( 1 + 13.9iT - 5.04e3T^{2} \)
73 \( 1 + 56.0T + 5.32e3T^{2} \)
79 \( 1 + 118. iT - 6.24e3T^{2} \)
83 \( 1 - 8.08T + 6.88e3T^{2} \)
89 \( 1 + 83.9T + 7.92e3T^{2} \)
97 \( 1 + 79.0T + 9.40e3T^{2} \)
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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

−12.25678304794253768305186648009, −11.56833795838567385140507389292, −9.774043800246973762658330521691, −9.092462361740834610209473899509, −8.483637230663851106645169734594, −7.05500671374323269216152265685, −6.20693303719328277158416224343, −4.96086674391386680186680621939, −3.97793736929304609167134001351, −1.29558982952261470886557313122, 1.13069121880165783905702721963, 3.03678885844550072327707838662, 3.87928815454972093082604776754, 5.49394935601108089786778619778, 6.97193515513747963834178470956, 8.011738289488228063806760365985, 9.297372377999241361169757263040, 10.13107392996335318351756433111, 10.94191430210544772405849885781, 11.77859168564225415623328066789

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