↑

Properties

Label 2-6e3-72.29-c2-0-16
Degree $2$
Conductor $216$
Sign $0.157 + 0.987i$
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  + (−1.99 + 0.171i)2-s + (3.94 − 0.681i)4-s + (−0.344 − 0.596i)5-s + (3.20 − 5.55i)7-s + (−7.73 + 2.03i)8-s + (0.788 + 1.13i)10-s + (2.32 − 4.03i)11-s + (−10.7 + 6.19i)13-s + (−5.43 + 11.6i)14-s + (15.0 − 5.37i)16-s − 26.4i·17-s + 11.2i·19-s + (−1.76 − 2.11i)20-s + (−3.94 + 8.42i)22-s + (1.52 − 0.882i)23-s + ⋯
L(s)  = 1  + (−0.996 + 0.0855i)2-s + (0.985 − 0.170i)4-s + (−0.0689 − 0.119i)5-s + (0.458 − 0.793i)7-s + (−0.967 + 0.254i)8-s + (0.0788 + 0.113i)10-s + (0.211 − 0.366i)11-s + (−0.825 + 0.476i)13-s + (−0.388 + 0.829i)14-s + (0.941 − 0.335i)16-s − 1.55i·17-s + 0.590i·19-s + (−0.0882 − 0.105i)20-s + (−0.179 + 0.383i)22-s + (0.0664 − 0.0383i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.157 + 0.987i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.157 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.157 + 0.987i$
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: \((2,\ 216,\ (\ :1),\ 0.157 + 0.987i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.650811 - 0.555287i\)
\(L(\frac12)\) \(\approx\) \(0.650811 - 0.555287i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.99 - 0.171i)T \)
3 \( 1 \)
good5 \( 1 + (0.344 + 0.596i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-3.20 + 5.55i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-2.32 + 4.03i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (10.7 - 6.19i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 26.4iT - 289T^{2} \)
19 \( 1 - 11.2iT - 361T^{2} \)
23 \( 1 + (-1.52 + 0.882i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-11.0 + 19.2i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (27.1 + 47.0i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 57.9iT - 1.36e3T^{2} \)
41 \( 1 + (-47.1 + 27.1i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-24.2 - 14.0i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-20.3 - 11.7i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 97.9T + 2.80e3T^{2} \)
59 \( 1 + (-38.4 - 66.6i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (0.493 + 0.284i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (58.9 - 34.0i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 59.1iT - 5.04e3T^{2} \)
73 \( 1 - 19.6T + 5.32e3T^{2} \)
79 \( 1 + (63.2 - 109. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-40.9 + 70.9i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 46.6iT - 7.92e3T^{2} \)
97 \( 1 + (-32.6 + 56.6i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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

−11.58541609527649025935121635124, −10.86622350836715319378572876796, −9.781606702042035971760745881313, −9.031885201812037183726267227325, −7.73930274163456032647184878150, −7.19254933780936346172263243351, −5.83089066928020086814245676879, −4.29929856949516600850374669123, −2.45510522907246910824496607031, −0.66228145878439984334251001939, 1.68374639394022978419498129652, 3.11435752681595414179504346631, 5.06827827171692157319907354863, 6.38482580773269435210331422866, 7.47349372415728685960366229610, 8.467924385538611051225634284914, 9.258163147957340270175182322984, 10.37483043690077883537198308790, 11.13591492073779733673092711740, 12.22037221053715978322389424794

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