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Properties

Label 2-162-9.7-c1-0-1
Degree $2$
Conductor $162$
Sign $0.766 + 0.642i$
Analytic cond. $1.29357$
Root an. cond. $1.13735$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (1.5 + 2.59i)5-s + (2 − 3.46i)7-s − 0.999·8-s + 3·10-s + (0.5 + 0.866i)13-s + (−1.99 − 3.46i)14-s + (−0.5 + 0.866i)16-s − 3·17-s − 4·19-s + (1.50 − 2.59i)20-s + (−2 + 3.46i)25-s + 0.999·26-s − 3.99·28-s + (−4.5 + 7.79i)29-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.670 + 1.16i)5-s + (0.755 − 1.30i)7-s − 0.353·8-s + 0.948·10-s + (0.138 + 0.240i)13-s + (−0.534 − 0.925i)14-s + (−0.125 + 0.216i)16-s − 0.727·17-s − 0.917·19-s + (0.335 − 0.580i)20-s + (−0.400 + 0.692i)25-s + 0.196·26-s − 0.755·28-s + (−0.835 + 1.44i)29-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :1/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37678 - 0.501109i\)
\(L(\frac12)\) \(\approx\) \(1.37678 - 0.501109i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
good5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2 + 3.46i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.5 - 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6 + 10.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 + (-8 + 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.93429128948187488972994131825, −11.48119090564304142399255937314, −10.66866828775986187375433701753, −10.28413299592671584601539887481, −8.820286420381976713018688796114, −7.27722457938436981513682693788, −6.40240599949394411915951835523, −4.81645177201045047128269588789, −3.53881614318847338394124695872, −1.91882084049818752724148443255, 2.15987477796209772498727713910, 4.45277659599994493652351858628, 5.41192272481053456547808198414, 6.23401038195019494050085573043, 8.021285306323536167733686041875, 8.739907206134921971723465010490, 9.550142578397534869870381570351, 11.22792669909327103848765277347, 12.26625879428020869388346597525, 13.01340994152116054431240305845

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