Properties

Label 2-162-27.11-c2-0-5
Degree $2$
Conductor $162$
Sign $-0.200 + 0.979i$
Analytic cond. $4.41418$
Root an. cond. $2.10099$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.909 − 1.08i)2-s + (−0.347 − 1.96i)4-s + (1.54 − 4.23i)5-s + (0.223 − 1.26i)7-s + (−2.44 − 1.41i)8-s + (−3.18 − 5.51i)10-s + (−2.33 − 6.41i)11-s + (8.48 − 7.11i)13-s + (−1.17 − 1.39i)14-s + (−3.75 + 1.36i)16-s + (−24.1 + 13.9i)17-s + (14.5 − 25.1i)19-s + (−8.87 − 1.56i)20-s + (−9.07 − 3.30i)22-s + (29.2 − 5.15i)23-s + ⋯
L(s)  = 1  + (0.454 − 0.541i)2-s + (−0.0868 − 0.492i)4-s + (0.308 − 0.846i)5-s + (0.0319 − 0.181i)7-s + (−0.306 − 0.176i)8-s + (−0.318 − 0.551i)10-s + (−0.212 − 0.583i)11-s + (0.652 − 0.547i)13-s + (−0.0836 − 0.0997i)14-s + (−0.234 + 0.0855i)16-s + (−1.42 + 0.821i)17-s + (0.764 − 1.32i)19-s + (−0.443 − 0.0782i)20-s + (−0.412 − 0.150i)22-s + (1.27 − 0.223i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.200 + 0.979i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.200 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $-0.200 + 0.979i$
Analytic conductor: \(4.41418\)
Root analytic conductor: \(2.10099\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :1),\ -0.200 + 0.979i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.15733 - 1.41812i\)
\(L(\frac12)\) \(\approx\) \(1.15733 - 1.41812i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.909 + 1.08i)T \)
3 \( 1 \)
good5 \( 1 + (-1.54 + 4.23i)T + (-19.1 - 16.0i)T^{2} \)
7 \( 1 + (-0.223 + 1.26i)T + (-46.0 - 16.7i)T^{2} \)
11 \( 1 + (2.33 + 6.41i)T + (-92.6 + 77.7i)T^{2} \)
13 \( 1 + (-8.48 + 7.11i)T + (29.3 - 166. i)T^{2} \)
17 \( 1 + (24.1 - 13.9i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-14.5 + 25.1i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-29.2 + 5.15i)T + (497. - 180. i)T^{2} \)
29 \( 1 + (9.87 - 11.7i)T + (-146. - 828. i)T^{2} \)
31 \( 1 + (-8.09 - 45.8i)T + (-903. + 328. i)T^{2} \)
37 \( 1 + (6.62 + 11.4i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-27.2 - 32.4i)T + (-291. + 1.65e3i)T^{2} \)
43 \( 1 + (48.0 - 17.4i)T + (1.41e3 - 1.18e3i)T^{2} \)
47 \( 1 + (-57.3 - 10.1i)T + (2.07e3 + 755. i)T^{2} \)
53 \( 1 - 72.9iT - 2.80e3T^{2} \)
59 \( 1 + (-10.4 + 28.6i)T + (-2.66e3 - 2.23e3i)T^{2} \)
61 \( 1 + (-2.62 + 14.9i)T + (-3.49e3 - 1.27e3i)T^{2} \)
67 \( 1 + (-86.2 + 72.4i)T + (779. - 4.42e3i)T^{2} \)
71 \( 1 + (7.53 - 4.34i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (52.2 - 90.5i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (46.1 + 38.7i)T + (1.08e3 + 6.14e3i)T^{2} \)
83 \( 1 + (-17.6 + 21.0i)T + (-1.19e3 - 6.78e3i)T^{2} \)
89 \( 1 + (106. + 61.5i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-40.5 + 14.7i)T + (7.20e3 - 6.04e3i)T^{2} \)
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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

−12.64547004553376474576250982011, −11.19994370726809807685967553990, −10.69600724052190189006843937866, −9.181400233926807321589741182016, −8.577460419305248154858899996956, −6.86018193431710553252124748461, −5.52959106100290831005020113723, −4.55810758219926272619914488747, −3.00166340660497258575892998672, −1.10024339983651028230053538124, 2.43920673579725196039087652153, 3.98926552261320731905928676041, 5.41394986725472946803124645364, 6.58944997600149482971832744802, 7.36821741667391670689601950825, 8.745166466203171879404283247150, 9.841957336216179970543054932008, 11.05908749083994920520028082401, 11.92000127407037011539432075630, 13.22414067386678560269593055221

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