Properties

Label 2-6e3-216.11-c1-0-7
Degree $2$
Conductor $216$
Sign $0.0335 - 0.999i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.505 + 1.32i)2-s + (−1.38 − 1.04i)3-s + (−1.48 + 1.33i)4-s + (3.21 + 1.16i)5-s + (0.683 − 2.35i)6-s + (−0.199 − 0.0351i)7-s + (−2.51 − 1.28i)8-s + (0.811 + 2.88i)9-s + (0.0809 + 4.83i)10-s + (1.37 + 3.78i)11-s + (3.45 − 0.287i)12-s + (1.09 + 1.30i)13-s + (−0.0543 − 0.280i)14-s + (−3.21 − 4.97i)15-s + (0.429 − 3.97i)16-s + (1.57 − 0.908i)17-s + ⋯
L(s)  = 1  + (0.357 + 0.933i)2-s + (−0.796 − 0.603i)3-s + (−0.744 + 0.668i)4-s + (1.43 + 0.523i)5-s + (0.278 − 0.960i)6-s + (−0.0752 − 0.0132i)7-s + (−0.890 − 0.455i)8-s + (0.270 + 0.962i)9-s + (0.0256 + 1.52i)10-s + (0.415 + 1.14i)11-s + (0.996 − 0.0830i)12-s + (0.304 + 0.362i)13-s + (−0.0145 − 0.0750i)14-s + (−0.829 − 1.28i)15-s + (0.107 − 0.994i)16-s + (0.381 − 0.220i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0335 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0335 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.0335 - 0.999i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ 0.0335 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.916436 + 0.886179i\)
\(L(\frac12)\) \(\approx\) \(0.916436 + 0.886179i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.505 - 1.32i)T \)
3 \( 1 + (1.38 + 1.04i)T \)
good5 \( 1 + (-3.21 - 1.16i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (0.199 + 0.0351i)T + (6.57 + 2.39i)T^{2} \)
11 \( 1 + (-1.37 - 3.78i)T + (-8.42 + 7.07i)T^{2} \)
13 \( 1 + (-1.09 - 1.30i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (-1.57 + 0.908i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.24 - 5.61i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.37 + 7.80i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (3.08 + 2.58i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-7.51 + 1.32i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-1.53 + 0.885i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.61 + 7.88i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (-3.47 + 1.26i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-0.523 + 2.97i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 - 9.46T + 53T^{2} \)
59 \( 1 + (-1.23 + 3.38i)T + (-45.1 - 37.9i)T^{2} \)
61 \( 1 + (8.13 + 1.43i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (1.26 - 1.06i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (2.24 + 3.89i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.05 - 7.01i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.380 - 0.453i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (8.02 - 9.56i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (-10.7 - 6.18i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.24 - 3.00i)T + (74.3 - 62.3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.65427125159639852975327611601, −11.99944253080580066080059195920, −10.40831553223473454883388827493, −9.746931334843489734890960621045, −8.346750747816881721404391571078, −7.03866634937297699042591584203, −6.38031552399791718929492279099, −5.65550170572229590337763422151, −4.36574113592990662468473401352, −2.12539866483921566273604661620, 1.24663278940406614463887537537, 3.19628530157400930238019333424, 4.67459915954538015926002674679, 5.68360129172146169813012904636, 6.25195413504208993094813299766, 8.725964604686952390031679469151, 9.450192898662315190813423191815, 10.24559515017747698438192369442, 11.11610723496350367084124507560, 11.91972180935668755897742498535

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