↑

Properties

Label 2-162-9.4-c9-0-30
Degree $2$
Conductor $162$
Sign $0.642 + 0.766i$
Analytic cond. $83.4358$
Root an. cond. $9.13432$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (8 + 13.8i)2-s + (−127. + 221. i)4-s + (646. − 1.12e3i)5-s + (1.36e3 + 2.36e3i)7-s − 4.09e3·8-s + 2.07e4·10-s + (3.43e3 + 5.94e3i)11-s + (6.23e4 − 1.08e5i)13-s + (−2.18e4 + 3.78e4i)14-s + (−3.27e4 − 5.67e4i)16-s + 2.03e5·17-s − 8.48e5·19-s + (1.65e5 + 2.86e5i)20-s + (−5.49e4 + 9.51e4i)22-s + (2.62e5 − 4.54e5i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.462 − 0.801i)5-s + (0.215 + 0.372i)7-s − 0.353·8-s + 0.654·10-s + (0.0707 + 0.122i)11-s + (0.605 − 1.04i)13-s + (−0.152 + 0.263i)14-s + (−0.125 − 0.216i)16-s + 0.591·17-s − 1.49·19-s + (0.231 + 0.400i)20-s + (−0.0500 + 0.0866i)22-s + (0.195 − 0.338i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(83.4358\)
Root analytic conductor: \(9.13432\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :9/2),\ 0.642 + 0.766i)\)

Particular Values

\(L(5)\) \(\approx\) \(2.265338730\)
\(L(\frac12)\) \(\approx\) \(2.265338730\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-8 - 13.8i)T \)
3 \( 1 \)
good5 \( 1 + (-646. + 1.12e3i)T + (-9.76e5 - 1.69e6i)T^{2} \)
7 \( 1 + (-1.36e3 - 2.36e3i)T + (-2.01e7 + 3.49e7i)T^{2} \)
11 \( 1 + (-3.43e3 - 5.94e3i)T + (-1.17e9 + 2.04e9i)T^{2} \)
13 \( 1 + (-6.23e4 + 1.08e5i)T + (-5.30e9 - 9.18e9i)T^{2} \)
17 \( 1 - 2.03e5T + 1.18e11T^{2} \)
19 \( 1 + 8.48e5T + 3.22e11T^{2} \)
23 \( 1 + (-2.62e5 + 4.54e5i)T + (-9.00e11 - 1.55e12i)T^{2} \)
29 \( 1 + (-1.94e5 - 3.36e5i)T + (-7.25e12 + 1.25e13i)T^{2} \)
31 \( 1 + (-1.22e6 + 2.12e6i)T + (-1.32e13 - 2.28e13i)T^{2} \)
37 \( 1 + 8.77e6T + 1.29e14T^{2} \)
41 \( 1 + (2.20e6 - 3.81e6i)T + (-1.63e14 - 2.83e14i)T^{2} \)
43 \( 1 + (6.84e6 + 1.18e7i)T + (-2.51e14 + 4.35e14i)T^{2} \)
47 \( 1 + (1.50e7 + 2.59e7i)T + (-5.59e14 + 9.69e14i)T^{2} \)
53 \( 1 + 1.07e8T + 3.29e15T^{2} \)
59 \( 1 + (-6.32e7 + 1.09e8i)T + (-4.33e15 - 7.50e15i)T^{2} \)
61 \( 1 + (-3.68e6 - 6.37e6i)T + (-5.84e15 + 1.01e16i)T^{2} \)
67 \( 1 + (-1.26e8 + 2.19e8i)T + (-1.36e16 - 2.35e16i)T^{2} \)
71 \( 1 - 2.69e8T + 4.58e16T^{2} \)
73 \( 1 - 2.16e7T + 5.88e16T^{2} \)
79 \( 1 + (-1.01e8 - 1.74e8i)T + (-5.99e16 + 1.03e17i)T^{2} \)
83 \( 1 + (5.49e7 + 9.51e7i)T + (-9.34e16 + 1.61e17i)T^{2} \)
89 \( 1 - 2.44e8T + 3.50e17T^{2} \)
97 \( 1 + (4.56e8 + 7.90e8i)T + (-3.80e17 + 6.58e17i)T^{2} \)
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

−11.06158341496496927114851436650, −9.873178581045350036176515076766, −8.682990082896042543196025322951, −8.115895821953675101971225058261, −6.63296388745093918966801257009, −5.60988768161083185580708442675, −4.81670743167290672990993434312, −3.44494996928361724288330296923, −1.89089694090311429217336964429, −0.46166164448600542165223358902, 1.23637038724789823419697216540, 2.31032269923550871309095072180, 3.53833232525003764810777157716, 4.61434044404467118926422061990, 6.06989914387655667609068832294, 6.86527166250863000044800309363, 8.374079560534411441289487564379, 9.529007996099651757745592996957, 10.56031587456684938572629030544, 11.13183845446991340856969566772

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