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Properties

Label 2-6e3-72.59-c1-0-8
Degree $2$
Conductor $216$
Sign $0.569 + 0.821i$
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  + (1.22 − 0.707i)2-s + (0.999 − 1.73i)4-s − 2.82i·8-s + (3.27 − 1.89i)11-s + (−2.00 − 3.46i)16-s + 8.02i·17-s − 8.34·19-s + (2.67 − 4.63i)22-s + (2.5 + 4.33i)25-s + (−4.89 − 2.82i)32-s + (5.67 + 9.82i)34-s + (−10.2 + 5.90i)38-s + (0.398 + 0.230i)41-s + (1.17 + 2.03i)43-s − 7.56i·44-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)2-s + (0.499 − 0.866i)4-s − 0.999i·8-s + (0.987 − 0.570i)11-s + (−0.500 − 0.866i)16-s + 1.94i·17-s − 1.91·19-s + (0.570 − 0.987i)22-s + (0.5 + 0.866i)25-s + (−0.866 − 0.499i)32-s + (0.973 + 1.68i)34-s + (−1.65 + 0.957i)38-s + (0.0623 + 0.0359i)41-s + (0.179 + 0.310i)43-s − 1.14i·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.569 + 0.821i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ 0.569 + 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70596 - 0.893077i\)
\(L(\frac12)\) \(\approx\) \(1.70596 - 0.893077i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.707i)T \)
3 \( 1 \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.27 + 1.89i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 8.02iT - 17T^{2} \)
19 \( 1 + 8.34T + 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (-0.398 - 0.230i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.17 - 2.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (10.6 + 6.13i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.17 + 12.4i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 13.6T + 73T^{2} \)
79 \( 1 + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.44 + 1.41i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 5.65iT - 89T^{2} \)
97 \( 1 + (9.84 + 17.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.43456144736014910415493009798, −11.11505880538692027962030171053, −10.67628323460185008589191173847, −9.369287990429605082656202822531, −8.275978594695943027842292211063, −6.60744639700106978801550474519, −5.97847205754798626389243813262, −4.45271422773790686402333866509, −3.49579754885101005064022995249, −1.74412691181042305409975848809, 2.46302574536800100006982858466, 4.05003183004130308464428154322, 4.98389775598859159033844798926, 6.41175360277737464346448434713, 7.08122038014450191428570471897, 8.355449515447404044309301849598, 9.386027875713310378169847518964, 10.78394701103203452283123766908, 11.81735880985357457280207076333, 12.47625318556484993912200283553

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