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Properties

Label 2-162-9.4-c3-0-9
Degree $2$
Conductor $162$
Sign $0.642 + 0.766i$
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 + (5.59 − 9.69i)5-s + (−9.19 − 15.9i)7-s − 7.99·8-s + 22.3·10-s + (−11.7 − 20.4i)11-s + (33.8 − 58.6i)13-s + (18.3 − 31.8i)14-s + (−8 − 13.8i)16-s − 117.·17-s + 110.·19-s + (22.3 + 38.7i)20-s + (23.5 − 40.8i)22-s + (34.6 − 59.9i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.500 − 0.867i)5-s + (−0.496 − 0.860i)7-s − 0.353·8-s + 0.708·10-s + (−0.323 − 0.559i)11-s + (0.722 − 1.25i)13-s + (0.351 − 0.608i)14-s + (−0.125 − 0.216i)16-s − 1.67·17-s + 1.33·19-s + (0.250 + 0.433i)20-s + (0.228 − 0.395i)22-s + (0.313 − 0.543i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(9.55830\)
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: \((2,\ 162,\ (\ :3/2),\ 0.642 + 0.766i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.59374 - 0.743176i\)
\(L(\frac12)\) \(\approx\) \(1.59374 - 0.743176i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.59 + 9.69i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (9.19 + 15.9i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (11.7 + 20.4i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-33.8 + 58.6i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 117.T + 4.91e3T^{2} \)
19 \( 1 - 110.T + 6.85e3T^{2} \)
23 \( 1 + (-34.6 + 59.9i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-99.1 - 171. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-155. + 269. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 206.T + 5.06e4T^{2} \)
41 \( 1 + (66.3 - 114. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-167. - 290. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (189. + 328. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 190.T + 1.48e5T^{2} \)
59 \( 1 + (168. - 292. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (138. + 240. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (332. - 575. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 528.T + 3.57e5T^{2} \)
73 \( 1 + 73.8T + 3.89e5T^{2} \)
79 \( 1 + (-239. - 415. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (89.8 + 155. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 846.T + 7.04e5T^{2} \)
97 \( 1 + (-336. - 582. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.77110875044862782119880420778, −11.26401264430716138666016889340, −10.17538725441507789439260765338, −8.998368759341429306530200994417, −8.105387781285197801906498311118, −6.82597914966550234224095616781, −5.72606297497477389862627349745, −4.65601391033279153591822417531, −3.19432041675979517994047283245, −0.74933473418558299744814902238, 1.99834062961252519655337517313, 3.10525931143210135911954977396, 4.70011602710937871464307076658, 6.12623507871714322945051037723, 6.92262715704729926057755687537, 8.803316782762476577126355291837, 9.602727933263857029366308128047, 10.64314225060835278800021598196, 11.57229916240699096280881285866, 12.42526840391554126567091272279

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