Properties

Label 2-162-9.5-c8-0-21
Degree $2$
Conductor $162$
Sign $0.939 + 0.342i$
Analytic cond. $65.9953$
Root an. cond. $8.12375$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−9.79 + 5.65i)2-s + (63.9 − 110. i)4-s + (202. + 116. i)5-s + (1.76e3 + 3.05e3i)7-s + 1.44e3i·8-s − 2.64e3·10-s + (1.74e4 − 1.00e4i)11-s + (2.09e4 − 3.62e4i)13-s + (−3.46e4 − 1.99e4i)14-s + (−8.19e3 − 1.41e4i)16-s − 9.47e4i·17-s − 3.63e4·19-s + (2.58e4 − 1.49e4i)20-s + (−1.14e5 + 1.97e5i)22-s + (3.58e5 + 2.06e5i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.323 + 0.186i)5-s + (0.735 + 1.27i)7-s + 0.353i·8-s − 0.264·10-s + (1.19 − 0.689i)11-s + (0.732 − 1.26i)13-s + (−0.900 − 0.520i)14-s + (−0.125 − 0.216i)16-s − 1.13i·17-s − 0.278·19-s + (0.161 − 0.0933i)20-s + (−0.487 + 0.843i)22-s + (1.28 + 0.739i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(65.9953\)
Root analytic conductor: \(8.12375\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :4),\ 0.939 + 0.342i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.965460984\)
\(L(\frac12)\) \(\approx\) \(1.965460984\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (9.79 - 5.65i)T \)
3 \( 1 \)
good5 \( 1 + (-202. - 116. i)T + (1.95e5 + 3.38e5i)T^{2} \)
7 \( 1 + (-1.76e3 - 3.05e3i)T + (-2.88e6 + 4.99e6i)T^{2} \)
11 \( 1 + (-1.74e4 + 1.00e4i)T + (1.07e8 - 1.85e8i)T^{2} \)
13 \( 1 + (-2.09e4 + 3.62e4i)T + (-4.07e8 - 7.06e8i)T^{2} \)
17 \( 1 + 9.47e4iT - 6.97e9T^{2} \)
19 \( 1 + 3.63e4T + 1.69e10T^{2} \)
23 \( 1 + (-3.58e5 - 2.06e5i)T + (3.91e10 + 6.78e10i)T^{2} \)
29 \( 1 + (2.33e5 - 1.34e5i)T + (2.50e11 - 4.33e11i)T^{2} \)
31 \( 1 + (-2.35e5 + 4.08e5i)T + (-4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + 3.00e6T + 3.51e12T^{2} \)
41 \( 1 + (1.48e6 + 8.57e5i)T + (3.99e12 + 6.91e12i)T^{2} \)
43 \( 1 + (1.81e6 + 3.13e6i)T + (-5.84e12 + 1.01e13i)T^{2} \)
47 \( 1 + (-5.20e6 + 3.00e6i)T + (1.19e13 - 2.06e13i)T^{2} \)
53 \( 1 + 1.02e7iT - 6.22e13T^{2} \)
59 \( 1 + (-2.32e6 - 1.34e6i)T + (7.34e13 + 1.27e14i)T^{2} \)
61 \( 1 + (-2.72e6 - 4.71e6i)T + (-9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (-3.06e6 + 5.30e6i)T + (-2.03e14 - 3.51e14i)T^{2} \)
71 \( 1 - 2.11e7iT - 6.45e14T^{2} \)
73 \( 1 + 4.90e7T + 8.06e14T^{2} \)
79 \( 1 + (4.17e6 + 7.23e6i)T + (-7.58e14 + 1.31e15i)T^{2} \)
83 \( 1 + (-4.45e7 + 2.56e7i)T + (1.12e15 - 1.95e15i)T^{2} \)
89 \( 1 + 1.07e8iT - 3.93e15T^{2} \)
97 \( 1 + (1.02e7 + 1.76e7i)T + (-3.91e15 + 6.78e15i)T^{2} \)
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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

−11.32711022284112812419126537154, −10.18983781472207189710881998783, −8.903772400814129608565655650663, −8.558851087038659883125610393106, −7.15741994428207052551993336596, −5.94027331688397822835962678751, −5.22260264765886095625149082589, −3.26835024369286242636142376726, −1.91500594348254091648637025320, −0.65645428640558675201804515367, 1.21752319812057277565268710492, 1.69494587604474494071060505663, 3.74768232879616526365292333773, 4.55497345522186496811041855296, 6.46297224512969415332632360089, 7.25022666923371153021473963175, 8.550212180620186268835662466230, 9.336323280888596561349966238054, 10.50571710751188575872329009591, 11.18114085700747958769969009009

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