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Properties

Label 2-162-9.4-c1-0-1
Degree $2$
Conductor $162$
Sign $-0.173 - 0.984i$
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 + (0.5 + 0.866i)7-s − 0.999·8-s − 3·10-s + (1.5 + 2.59i)11-s + (2 − 3.46i)13-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s + 2·19-s + (−1.50 − 2.59i)20-s + (−1.5 + 2.59i)22-s + (3 − 5.19i)23-s + (−2 − 3.46i)25-s + 3.99·26-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.670 + 1.16i)5-s + (0.188 + 0.327i)7-s − 0.353·8-s − 0.948·10-s + (0.452 + 0.783i)11-s + (0.554 − 0.960i)13-s + (−0.133 + 0.231i)14-s + (−0.125 − 0.216i)16-s + 0.458·19-s + (−0.335 − 0.580i)20-s + (−0.319 + 0.553i)22-s + (0.625 − 1.08i)23-s + (−0.400 − 0.692i)25-s + 0.784·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :1/2),\ -0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.781091 + 0.930869i\)
\(L(\frac12)\) \(\approx\) \(0.781091 + 0.930869i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5 - 8.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 9T + 53T^{2} \)
59 \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7 - 12.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.5 - 2.59i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 18T + 89T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.20502644880271505762667057528, −12.19283051970361002428092658245, −11.23613223905438751972022657379, −10.25053277570361930924207602017, −8.834936174781593452814909941403, −7.66905073689947661154738095313, −6.90330704735414953485190453403, −5.71641139191517235358585726381, −4.20445647602309845509273962127, −2.91754008419131711827962264885, 1.24707914124000999913542719082, 3.59206387325909758127408916542, 4.56099257260541310401616603002, 5.81731870596322255653774703433, 7.44846691449250731958547779257, 8.751635463279047721042983295697, 9.372554760033261947426602530755, 11.01121290786422321051583622792, 11.56723079676035538687377000221, 12.52957867937897606301988860266

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