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Properties

Label 2-162-9.4-c3-0-10
Degree $2$
Conductor $162$
Sign $-0.766 + 0.642i$
Analytic cond. $9.55830$
Root an. cond. $3.09165$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)2-s + (−1.99 + 3.46i)4-s + (10.5 − 18.1i)5-s + (−4 − 6.92i)7-s + 7.99·8-s − 42·10-s + (18 + 31.1i)11-s + (24.5 − 42.4i)13-s + (−7.99 + 13.8i)14-s + (−8 − 13.8i)16-s − 21·17-s − 112·19-s + (42 + 72.7i)20-s + (36 − 62.3i)22-s + (90 − 155. i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.939 − 1.62i)5-s + (−0.215 − 0.374i)7-s + 0.353·8-s − 1.32·10-s + (0.493 + 0.854i)11-s + (0.522 − 0.905i)13-s + (−0.152 + 0.264i)14-s + (−0.125 − 0.216i)16-s − 0.299·17-s − 1.35·19-s + (0.469 + 0.813i)20-s + (0.348 − 0.604i)22-s + (0.815 − 1.41i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $-0.766 + 0.642i$
Analytic conductor: \(9.55830\)
Root analytic conductor: \(3.09165\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :3/2),\ -0.766 + 0.642i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.479782 - 1.31819i\)
\(L(\frac12)\) \(\approx\) \(0.479782 - 1.31819i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + 1.73i)T \)
3 \( 1 \)
good5 \( 1 + (-10.5 + 18.1i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (4 + 6.92i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-18 - 31.1i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-24.5 + 42.4i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 21T + 4.91e3T^{2} \)
19 \( 1 + 112T + 6.85e3T^{2} \)
23 \( 1 + (-90 + 155. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (67.5 + 116. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (154 - 266. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + T + 5.06e4T^{2} \)
41 \( 1 + (21 - 36.3i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (10 + 17.3i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-42 - 72.7i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 174T + 1.48e5T^{2} \)
59 \( 1 + (-252 + 436. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-192.5 - 333. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (136 - 235. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 888T + 3.57e5T^{2} \)
73 \( 1 - 371T + 3.89e5T^{2} \)
79 \( 1 + (-326 - 564. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-42 - 72.7i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 21T + 7.04e5T^{2} \)
97 \( 1 + (-623 - 1.07e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.37365569412686385313093478142, −10.79647128807278007024727004397, −9.951693221786258885633588973802, −8.972271204568524115182737871001, −8.344339804060036266096892806714, −6.63646743232648757435686047849, −5.19645933390127768821127762017, −4.13320893001743525914837271162, −2.07105184789143414197317796633, −0.75160116229901187165317760534, 2.03371887894989969365295511358, 3.61725175184557564453483206474, 5.74627633160831347732966629430, 6.41257781912142378158782433183, 7.27378153000945299767131004090, 8.871044944957653067071524676734, 9.575941390545464730437556384030, 10.83286639728749434244102639692, 11.32444565377788754001933627804, 13.19636071814474684010336587790

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