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Properties

Label 2-162-9.5-c6-0-16
Degree $2$
Conductor $162$
Sign $0.173 + 0.984i$
Analytic cond. $37.2687$
Root an. cond. $6.10481$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.89 + 2.82i)2-s + (15.9 − 27.7i)4-s + (190. + 109. i)5-s + (−180. − 313. i)7-s + 181. i·8-s − 1.24e3·10-s + (56.7 − 32.7i)11-s + (1.62e3 − 2.81e3i)13-s + (1.77e3 + 1.02e3i)14-s + (−512. − 886. i)16-s − 6.32e3i·17-s − 8.69e3·19-s + (6.09e3 − 3.51e3i)20-s + (−185. + 321. i)22-s + (−1.13e4 − 6.54e3i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (1.52 + 0.879i)5-s + (−0.527 − 0.913i)7-s + 0.353i·8-s − 1.24·10-s + (0.0426 − 0.0246i)11-s + (0.739 − 1.28i)13-s + (0.645 + 0.372i)14-s + (−0.125 − 0.216i)16-s − 1.28i·17-s − 1.26·19-s + (0.761 − 0.439i)20-s + (−0.0174 + 0.0301i)22-s + (−0.932 − 0.538i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(37.2687\)
Root analytic conductor: \(6.10481\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :3),\ 0.173 + 0.984i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.270223508\)
\(L(\frac12)\) \(\approx\) \(1.270223508\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4.89 - 2.82i)T \)
3 \( 1 \)
good5 \( 1 + (-190. - 109. i)T + (7.81e3 + 1.35e4i)T^{2} \)
7 \( 1 + (180. + 313. i)T + (-5.88e4 + 1.01e5i)T^{2} \)
11 \( 1 + (-56.7 + 32.7i)T + (8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 + (-1.62e3 + 2.81e3i)T + (-2.41e6 - 4.18e6i)T^{2} \)
17 \( 1 + 6.32e3iT - 2.41e7T^{2} \)
19 \( 1 + 8.69e3T + 4.70e7T^{2} \)
23 \( 1 + (1.13e4 + 6.54e3i)T + (7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + (6.57e3 - 3.79e3i)T + (2.97e8 - 5.15e8i)T^{2} \)
31 \( 1 + (1.88e4 - 3.26e4i)T + (-4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + 2.14e4T + 2.56e9T^{2} \)
41 \( 1 + (-1.78e4 - 1.03e4i)T + (2.37e9 + 4.11e9i)T^{2} \)
43 \( 1 + (6.12e4 + 1.06e5i)T + (-3.16e9 + 5.47e9i)T^{2} \)
47 \( 1 + (1.36e4 - 7.88e3i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + 2.41e5iT - 2.21e10T^{2} \)
59 \( 1 + (-6.44e4 - 3.71e4i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (9.68e4 + 1.67e5i)T + (-2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (-6.78e4 + 1.17e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + 2.62e5iT - 1.28e11T^{2} \)
73 \( 1 - 4.18e5T + 1.51e11T^{2} \)
79 \( 1 + (-3.41e5 - 5.90e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (5.14e5 - 2.97e5i)T + (1.63e11 - 2.83e11i)T^{2} \)
89 \( 1 + 6.52e5iT - 4.96e11T^{2} \)
97 \( 1 + (8.77e5 + 1.51e6i)T + (-4.16e11 + 7.21e11i)T^{2} \)
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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

−10.95370206143282055377005461191, −10.35801073713278638301568437729, −9.726124678000054544102752144617, −8.500610838643259687950460559464, −7.08090753404833566025017227348, −6.41564945995339540782206038824, −5.37178400815123863379768560159, −3.32493854493522359238758080689, −1.99000285295231285330517753250, −0.42357019529693825410030169697, 1.54046493613576078617673146730, 2.21852094005953603881247864648, 4.13262461719487822607315591648, 5.84322858552710448479319713119, 6.36048998685115051398935532232, 8.333414265477898897854918530532, 9.104747174127751688449538517683, 9.674769600412065045451362842962, 10.78723782202207388168637986757, 12.06386691329413199692037521089

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