↑

Properties

Label 2-162-9.2-c6-0-16
Degree $2$
Conductor $162$
Sign $0.0871 + 0.996i$
Analytic cond. $37.2687$
Root an. cond. $6.10481$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−4.89 − 2.82i)2-s + (15.9 + 27.7i)4-s + (148. − 85.5i)5-s + (−280. + 485. i)7-s − 181. i·8-s − 967.·10-s + (190. + 110. i)11-s + (−1.89e3 − 3.27e3i)13-s + (2.74e3 − 1.58e3i)14-s + (−512. + 886. i)16-s − 2.66e3i·17-s + 1.03e4·19-s + (4.74e3 + 2.73e3i)20-s + (−623. − 1.07e3i)22-s + (−1.28e4 + 7.39e3i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (1.18 − 0.684i)5-s + (−0.817 + 1.41i)7-s − 0.353i·8-s − 0.967·10-s + (0.143 + 0.0827i)11-s + (−0.860 − 1.49i)13-s + (1.00 − 0.578i)14-s + (−0.125 + 0.216i)16-s − 0.541i·17-s + 1.50·19-s + (0.592 + 0.342i)20-s + (−0.0585 − 0.101i)22-s + (−1.05 + 0.607i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0871 + 0.996i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.0871 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.0871 + 0.996i$
Analytic conductor: \(37.2687\)
Root analytic conductor: \(6.10481\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :3),\ 0.0871 + 0.996i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.401196683\)
\(L(\frac12)\) \(\approx\) \(1.401196683\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4.89 + 2.82i)T \)
3 \( 1 \)
good5 \( 1 + (-148. + 85.5i)T + (7.81e3 - 1.35e4i)T^{2} \)
7 \( 1 + (280. - 485. i)T + (-5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + (-190. - 110. i)T + (8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 + (1.89e3 + 3.27e3i)T + (-2.41e6 + 4.18e6i)T^{2} \)
17 \( 1 + 2.66e3iT - 2.41e7T^{2} \)
19 \( 1 - 1.03e4T + 4.70e7T^{2} \)
23 \( 1 + (1.28e4 - 7.39e3i)T + (7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (-1.28e4 - 7.43e3i)T + (2.97e8 + 5.15e8i)T^{2} \)
31 \( 1 + (7.91e3 + 1.37e4i)T + (-4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 - 2.65e4T + 2.56e9T^{2} \)
41 \( 1 + (-9.86e4 + 5.69e4i)T + (2.37e9 - 4.11e9i)T^{2} \)
43 \( 1 + (-2.60e4 + 4.51e4i)T + (-3.16e9 - 5.47e9i)T^{2} \)
47 \( 1 + (-1.00e4 - 5.77e3i)T + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 + 1.37e5iT - 2.21e10T^{2} \)
59 \( 1 + (-2.52e5 + 1.45e5i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (-1.41e5 + 2.44e5i)T + (-2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (1.88e5 + 3.27e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 - 5.21e5iT - 1.28e11T^{2} \)
73 \( 1 + 6.11e5T + 1.51e11T^{2} \)
79 \( 1 + (-8.77e4 + 1.52e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + (3.14e5 + 1.81e5i)T + (1.63e11 + 2.83e11i)T^{2} \)
89 \( 1 + 1.23e6iT - 4.96e11T^{2} \)
97 \( 1 + (-6.24e5 + 1.08e6i)T + (-4.16e11 - 7.21e11i)T^{2} \)
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

−11.66289704865411442017186301345, −9.960405457363717202212380168274, −9.657688925138090706953248735520, −8.774149617523013833431140908634, −7.51078009524821716532191208865, −5.88929525609072708408990041885, −5.31543283191025476754669240164, −3.05738486702564141476691378617, −2.10423559074695765176991569975, −0.54847233948898516086311336804, 1.14412784467035063000242543253, 2.57958727438155529839058974646, 4.23145199163846965710569631730, 5.98990623428508437061968535287, 6.77104635975732766549969831654, 7.55238684788502190912420485807, 9.284204843126808698900036304333, 9.913514279479991390790204419061, 10.53329317223711435602822453462, 11.80660862013727661775739061144

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