Properties

Label 2-6e3-216.101-c2-0-19
Degree $2$
Conductor $216$
Sign $-0.866 - 0.499i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.785 + 1.83i)2-s + (2.34 + 1.87i)3-s + (−2.76 + 2.88i)4-s + (−0.130 − 0.741i)5-s + (−1.60 + 5.78i)6-s + (−1.32 − 1.11i)7-s + (−7.48 − 2.82i)8-s + (1.97 + 8.78i)9-s + (1.26 − 0.822i)10-s + (−2.40 + 13.6i)11-s + (−11.8 + 1.58i)12-s + (−5.13 + 14.1i)13-s + (1.00 − 3.30i)14-s + (1.08 − 1.98i)15-s + (−0.685 − 15.9i)16-s + (7.61 − 4.39i)17-s + ⋯
L(s)  = 1  + (0.392 + 0.919i)2-s + (0.780 + 0.624i)3-s + (−0.691 + 0.722i)4-s + (−0.0261 − 0.148i)5-s + (−0.267 + 0.963i)6-s + (−0.189 − 0.158i)7-s + (−0.935 − 0.352i)8-s + (0.219 + 0.975i)9-s + (0.126 − 0.0822i)10-s + (−0.218 + 1.24i)11-s + (−0.991 + 0.131i)12-s + (−0.395 + 1.08i)13-s + (0.0717 − 0.236i)14-s + (0.0722 − 0.132i)15-s + (−0.0428 − 0.999i)16-s + (0.448 − 0.258i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.866 - 0.499i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ -0.866 - 0.499i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.519808 + 1.94237i\)
\(L(\frac12)\) \(\approx\) \(0.519808 + 1.94237i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.785 - 1.83i)T \)
3 \( 1 + (-2.34 - 1.87i)T \)
good5 \( 1 + (0.130 + 0.741i)T + (-23.4 + 8.55i)T^{2} \)
7 \( 1 + (1.32 + 1.11i)T + (8.50 + 48.2i)T^{2} \)
11 \( 1 + (2.40 - 13.6i)T + (-113. - 41.3i)T^{2} \)
13 \( 1 + (5.13 - 14.1i)T + (-129. - 108. i)T^{2} \)
17 \( 1 + (-7.61 + 4.39i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-7.40 - 4.27i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (16.0 + 19.0i)T + (-91.8 + 520. i)T^{2} \)
29 \( 1 + (-31.3 + 11.4i)T + (644. - 540. i)T^{2} \)
31 \( 1 + (-33.4 + 28.0i)T + (166. - 946. i)T^{2} \)
37 \( 1 + (-26.0 + 15.0i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (16.2 - 44.7i)T + (-1.28e3 - 1.08e3i)T^{2} \)
43 \( 1 + (-24.9 - 4.39i)T + (1.73e3 + 632. i)T^{2} \)
47 \( 1 + (31.4 - 37.4i)T + (-383. - 2.17e3i)T^{2} \)
53 \( 1 + 16.3T + 2.80e3T^{2} \)
59 \( 1 + (-6.61 - 37.5i)T + (-3.27e3 + 1.19e3i)T^{2} \)
61 \( 1 + (-39.1 + 46.6i)T + (-646. - 3.66e3i)T^{2} \)
67 \( 1 + (40.7 - 112. i)T + (-3.43e3 - 2.88e3i)T^{2} \)
71 \( 1 + (36.2 - 20.9i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-52.5 + 91.0i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-31.0 + 11.3i)T + (4.78e3 - 4.01e3i)T^{2} \)
83 \( 1 + (-70.8 + 25.7i)T + (5.27e3 - 4.42e3i)T^{2} \)
89 \( 1 + (4.93 + 2.85i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-14.9 + 84.7i)T + (-8.84e3 - 3.21e3i)T^{2} \)
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

−12.72225223983769438203886889016, −11.80938516226222940908165471940, −10.05561870526329258165672459366, −9.530861170404599151770328750414, −8.377331500185700923200089114291, −7.52746309828403471771174919471, −6.45903824938517688407053581285, −4.80840950450125875051824701306, −4.25442285675265328020423015527, −2.65720012378816419287143182007, 0.967625071234178030704630692030, 2.77775772535821778177425629146, 3.45608776380617444207802369547, 5.25551022225385340335636751778, 6.39857818499122487512750389633, 7.930064634636795193286837949826, 8.747067616107424472838150919963, 9.852538787501028498697339499283, 10.75308130573892373419370146171, 11.93977991033440654579046064284

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