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Properties

Label 2-162-9.7-c7-0-5
Degree $2$
Conductor $162$
Sign $0.642 - 0.766i$
Analytic cond. $50.6063$
Root an. cond. $7.11381$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4 + 6.92i)2-s + (−31.9 − 55.4i)4-s + (−235. − 408. i)5-s + (−367. + 636. i)7-s + 511.·8-s + 3.77e3·10-s + (−2.66e3 + 4.60e3i)11-s + (−4.43e3 − 7.67e3i)13-s + (−2.94e3 − 5.09e3i)14-s + (−2.04e3 + 3.54e3i)16-s − 3.41e4·17-s − 8.70e3·19-s + (−1.50e4 + 2.61e4i)20-s + (−2.12e4 − 3.68e4i)22-s + (−2.60e4 − 4.51e4i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.843 − 1.46i)5-s + (−0.405 + 0.701i)7-s + 0.353·8-s + 1.19·10-s + (−0.602 + 1.04i)11-s + (−0.559 − 0.969i)13-s + (−0.286 − 0.496i)14-s + (−0.125 + 0.216i)16-s − 1.68·17-s − 0.291·19-s + (−0.421 + 0.730i)20-s + (−0.426 − 0.738i)22-s + (−0.446 − 0.773i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.642 - 0.766i$
Analytic conductor: \(50.6063\)
Root analytic conductor: \(7.11381\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :7/2),\ 0.642 - 0.766i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.5607354604\)
\(L(\frac12)\) \(\approx\) \(0.5607354604\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4 - 6.92i)T \)
3 \( 1 \)
good5 \( 1 + (235. + 408. i)T + (-3.90e4 + 6.76e4i)T^{2} \)
7 \( 1 + (367. - 636. i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + (2.66e3 - 4.60e3i)T + (-9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 + (4.43e3 + 7.67e3i)T + (-3.13e7 + 5.43e7i)T^{2} \)
17 \( 1 + 3.41e4T + 4.10e8T^{2} \)
19 \( 1 + 8.70e3T + 8.93e8T^{2} \)
23 \( 1 + (2.60e4 + 4.51e4i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 + (-1.06e5 + 1.84e5i)T + (-8.62e9 - 1.49e10i)T^{2} \)
31 \( 1 + (-1.16e5 - 2.01e5i)T + (-1.37e10 + 2.38e10i)T^{2} \)
37 \( 1 - 6.73e4T + 9.49e10T^{2} \)
41 \( 1 + (-2.25e4 - 3.90e4i)T + (-9.73e10 + 1.68e11i)T^{2} \)
43 \( 1 + (1.07e5 - 1.85e5i)T + (-1.35e11 - 2.35e11i)T^{2} \)
47 \( 1 + (-2.80e5 + 4.85e5i)T + (-2.53e11 - 4.38e11i)T^{2} \)
53 \( 1 + 2.44e5T + 1.17e12T^{2} \)
59 \( 1 + (9.66e5 + 1.67e6i)T + (-1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (1.06e6 - 1.84e6i)T + (-1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (-8.14e5 - 1.41e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 - 5.38e6T + 9.09e12T^{2} \)
73 \( 1 + 1.01e6T + 1.10e13T^{2} \)
79 \( 1 + (-3.87e6 + 6.70e6i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + (1.04e6 - 1.81e6i)T + (-1.35e13 - 2.35e13i)T^{2} \)
89 \( 1 - 1.17e7T + 4.42e13T^{2} \)
97 \( 1 + (9.61e5 - 1.66e6i)T + (-4.03e13 - 6.99e13i)T^{2} \)
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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

−12.00623627525008745674351045109, −10.45870156078439090842471292309, −9.370889438239395012661086394119, −8.484531124864597373201688008136, −7.80761514221843975512472991126, −6.45112814590231201236759327417, −5.05164377956970977674208037733, −4.42268968357139324854598370906, −2.34479546983453816364319512662, −0.55395323705862797919098087384, 0.31853878932003977321339357356, 2.34117534416299320622998616371, 3.36503305745231424375723216326, 4.35211546290866933134079740307, 6.45828164291577771189593479498, 7.23008229708142280760882464389, 8.291257550547796394814422162652, 9.586248279992620643019574331470, 10.80230844067287965269215245834, 11.03540444856480751740925215503

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