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Properties

Label 2-162-9.7-c9-0-32
Degree $2$
Conductor $162$
Sign $-0.939 + 0.342i$
Analytic cond. $83.4358$
Root an. cond. $9.13432$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−8 + 13.8i)2-s + (−127. − 221. i)4-s + (−435 − 753. i)5-s + (476 − 824. i)7-s + 4.09e3·8-s + 1.39e4·10-s + (2.80e4 − 4.86e4i)11-s + (−8.90e4 − 1.54e5i)13-s + (7.61e3 + 1.31e4i)14-s + (−3.27e4 + 5.67e4i)16-s − 2.47e5·17-s + 3.15e5·19-s + (−1.11e5 + 1.92e5i)20-s + (4.49e5 + 7.78e5i)22-s + (−1.02e5 − 1.77e5i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.311 − 0.539i)5-s + (0.0749 − 0.129i)7-s + 0.353·8-s + 0.440·10-s + (0.578 − 1.00i)11-s + (−0.864 − 1.49i)13-s + (0.0529 + 0.0917i)14-s + (−0.125 + 0.216i)16-s − 0.719·17-s + 0.555·19-s + (−0.155 + 0.269i)20-s + (0.408 + 0.708i)22-s + (−0.0761 − 0.131i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $-0.939 + 0.342i$
Analytic conductor: \(83.4358\)
Root analytic conductor: \(9.13432\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :9/2),\ -0.939 + 0.342i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.6678959148\)
\(L(\frac12)\) \(\approx\) \(0.6678959148\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (8 - 13.8i)T \)
3 \( 1 \)
good5 \( 1 + (435 + 753. i)T + (-9.76e5 + 1.69e6i)T^{2} \)
7 \( 1 + (-476 + 824. i)T + (-2.01e7 - 3.49e7i)T^{2} \)
11 \( 1 + (-2.80e4 + 4.86e4i)T + (-1.17e9 - 2.04e9i)T^{2} \)
13 \( 1 + (8.90e4 + 1.54e5i)T + (-5.30e9 + 9.18e9i)T^{2} \)
17 \( 1 + 2.47e5T + 1.18e11T^{2} \)
19 \( 1 - 3.15e5T + 3.22e11T^{2} \)
23 \( 1 + (1.02e5 + 1.77e5i)T + (-9.00e11 + 1.55e12i)T^{2} \)
29 \( 1 + (-1.92e6 + 3.32e6i)T + (-7.25e12 - 1.25e13i)T^{2} \)
31 \( 1 + (-6.54e5 - 1.13e6i)T + (-1.32e13 + 2.28e13i)T^{2} \)
37 \( 1 - 4.30e6T + 1.29e14T^{2} \)
41 \( 1 + (7.56e5 + 1.30e6i)T + (-1.63e14 + 2.83e14i)T^{2} \)
43 \( 1 + (1.68e7 - 2.91e7i)T + (-2.51e14 - 4.35e14i)T^{2} \)
47 \( 1 + (-5.29e6 + 9.16e6i)T + (-5.59e14 - 9.69e14i)T^{2} \)
53 \( 1 - 1.66e7T + 3.29e15T^{2} \)
59 \( 1 + (5.61e7 + 9.71e7i)T + (-4.33e15 + 7.50e15i)T^{2} \)
61 \( 1 + (-1.65e7 + 2.87e7i)T + (-5.84e15 - 1.01e16i)T^{2} \)
67 \( 1 + (-6.06e7 - 1.05e8i)T + (-1.36e16 + 2.35e16i)T^{2} \)
71 \( 1 + 3.87e8T + 4.58e16T^{2} \)
73 \( 1 - 2.55e8T + 5.88e16T^{2} \)
79 \( 1 + (2.46e8 - 4.26e8i)T + (-5.99e16 - 1.03e17i)T^{2} \)
83 \( 1 + (-2.28e8 + 3.96e8i)T + (-9.34e16 - 1.61e17i)T^{2} \)
89 \( 1 + 3.18e7T + 3.50e17T^{2} \)
97 \( 1 + (-3.36e8 + 5.83e8i)T + (-3.80e17 - 6.58e17i)T^{2} \)
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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

−10.57988663322676913665785370772, −9.552167270638106311525394859276, −8.457306209923211588869671144318, −7.81617423914516531894557493136, −6.52796391555725148667006725522, −5.44128610419513505449664148497, −4.36892134348968473739454488772, −2.86996092897739428115678814173, −1.01661319344092732578391792401, −0.20204467513050155024171218852, 1.51515986189496776702302930027, 2.50478634252530627421851677641, 3.88803369608805795454145471938, 4.87698346552909634861670500894, 6.76134582621943394800244861428, 7.35872476887102815344393405587, 8.867053828290336048503856375896, 9.583502587939571103210191951444, 10.63316754859130191429382468223, 11.72785004790774023299234776713

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