↑

Properties

Label 2-6e3-72.29-c2-0-12
Degree $2$
Conductor $216$
Sign $0.959 - 0.281i$
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  + (−0.310 + 1.97i)2-s + (−3.80 − 1.22i)4-s + (0.661 + 1.14i)5-s + (4.89 − 8.47i)7-s + (3.60 − 7.14i)8-s + (−2.47 + 0.952i)10-s + (6.81 − 11.8i)11-s + (−1.13 + 0.654i)13-s + (15.2 + 12.2i)14-s + (12.9 + 9.33i)16-s − 0.636i·17-s + 22.9i·19-s + (−1.11 − 5.17i)20-s + (21.2 + 17.1i)22-s + (22.3 − 12.9i)23-s + ⋯
L(s)  = 1  + (−0.155 + 0.987i)2-s + (−0.951 − 0.306i)4-s + (0.132 + 0.229i)5-s + (0.699 − 1.21i)7-s + (0.450 − 0.892i)8-s + (−0.247 + 0.0952i)10-s + (0.619 − 1.07i)11-s + (−0.0872 + 0.0503i)13-s + (1.08 + 0.878i)14-s + (0.812 + 0.583i)16-s − 0.0374i·17-s + 1.20i·19-s + (−0.0557 − 0.258i)20-s + (0.964 + 0.778i)22-s + (0.972 − 0.561i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.959 - 0.281i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ 0.959 - 0.281i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.40614 + 0.202158i\)
\(L(\frac12)\) \(\approx\) \(1.40614 + 0.202158i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.310 - 1.97i)T \)
3 \( 1 \)
good5 \( 1 + (-0.661 - 1.14i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-4.89 + 8.47i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-6.81 + 11.8i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (1.13 - 0.654i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 0.636iT - 289T^{2} \)
19 \( 1 - 22.9iT - 361T^{2} \)
23 \( 1 + (-22.3 + 12.9i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-6.64 + 11.5i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-18.7 - 32.5i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 51.3iT - 1.36e3T^{2} \)
41 \( 1 + (31.8 - 18.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (56.7 + 32.7i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-75.9 - 43.8i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 9.23T + 2.80e3T^{2} \)
59 \( 1 + (14.5 + 25.2i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (7.53 + 4.35i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-24.3 + 14.0i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 83.8iT - 5.04e3T^{2} \)
73 \( 1 + 88.8T + 5.32e3T^{2} \)
79 \( 1 + (-22.0 + 38.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (64.9 - 112. i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 23.0iT - 7.92e3T^{2} \)
97 \( 1 + (-1.36 + 2.36i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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

−12.23286238741051942957304342511, −10.84829700056141616764092779245, −10.23841238071456460658109675588, −8.887106758271731680266511494502, −8.068580398420175619026678336008, −7.04451994588681345565702979087, −6.13567891189922230037734991344, −4.80424038629937564867011450495, −3.68279197667270019821298628163, −0.990976358109658031950167973966, 1.55323701772352059148293734000, 2.82104744340768888759718251394, 4.55544245786590545649562755939, 5.34984465229091990243926920291, 7.11513521985257575556790648976, 8.527424796592967167206276606679, 9.136058200810299373281034126296, 10.05598156217197106309895126001, 11.39704892292114610766823453737, 11.82452858969805419426000893859

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