Properties

Label 2-162-9.2-c6-0-18
Degree $2$
Conductor $162$
Sign $0.939 - 0.342i$
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 + (146. − 84.8i)5-s + (−1 + 1.73i)7-s + 181. i·8-s + 960·10-s + (−29.3 − 16.9i)11-s + (1.47e3 + 2.55e3i)13-s + (−9.79 + 5.65i)14-s + (−512. + 886. i)16-s − 4.48e3i·17-s + 5.25e3·19-s + (4.70e3 + 2.71e3i)20-s + (−96 − 166. i)22-s + (8.87e3 − 5.12e3i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (1.17 − 0.678i)5-s + (−0.00291 + 0.00504i)7-s + 0.353i·8-s + 0.959·10-s + (−0.0220 − 0.0127i)11-s + (0.671 + 1.16i)13-s + (−0.00357 + 0.00206i)14-s + (−0.125 + 0.216i)16-s − 0.911i·17-s + 0.766·19-s + (0.587 + 0.339i)20-s + (−0.00901 − 0.0156i)22-s + (0.729 − 0.421i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(37.2687\)
Root analytic conductor: \(6.10481\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :3),\ 0.939 - 0.342i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(3.905191953\)
\(L(\frac12)\) \(\approx\) \(3.905191953\)
\(L(4)\) 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 \( 1 + (-146. + 84.8i)T + (7.81e3 - 1.35e4i)T^{2} \)
7 \( 1 + (1 - 1.73i)T + (-5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + (29.3 + 16.9i)T + (8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 + (-1.47e3 - 2.55e3i)T + (-2.41e6 + 4.18e6i)T^{2} \)
17 \( 1 + 4.48e3iT - 2.41e7T^{2} \)
19 \( 1 - 5.25e3T + 4.70e7T^{2} \)
23 \( 1 + (-8.87e3 + 5.12e3i)T + (7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (-1.91e3 - 1.10e3i)T + (2.97e8 + 5.15e8i)T^{2} \)
31 \( 1 + (1.14e4 + 1.98e4i)T + (-4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 - 3.40e4T + 2.56e9T^{2} \)
41 \( 1 + (-1.45e4 + 8.38e3i)T + (2.37e9 - 4.11e9i)T^{2} \)
43 \( 1 + (-3.20e3 + 5.54e3i)T + (-3.16e9 - 5.47e9i)T^{2} \)
47 \( 1 + (-1.55e5 - 8.99e4i)T + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 - 1.92e5iT - 2.21e10T^{2} \)
59 \( 1 + (-2.83e5 + 1.63e5i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (-3.12e4 + 5.41e4i)T + (-2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (2.19e5 + 3.79e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 - 6.82e4iT - 1.28e11T^{2} \)
73 \( 1 + 7.30e5T + 1.51e11T^{2} \)
79 \( 1 + (1.70e5 - 2.94e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + (-4.29e5 - 2.48e5i)T + (1.63e11 + 2.83e11i)T^{2} \)
89 \( 1 - 3.86e5iT - 4.96e11T^{2} \)
97 \( 1 + (-1.40e5 + 2.43e5i)T + (-4.16e11 - 7.21e11i)T^{2} \)
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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

−11.99434306903762877615806916579, −10.95922794742768957970359184882, −9.496833191588734921759721070073, −8.929946528673196166724919607536, −7.41322192915653128069404378863, −6.25437516295783102369798934076, −5.35409283033718394134411905096, −4.26712257480328775783731850027, −2.58120050985987963737727057501, −1.19628588494882062098064177755, 1.18935927989187356768736107792, 2.52920982396873063295103589816, 3.61673783899774643682870978292, 5.36914134116636138594873073585, 6.04084209406382272590562337587, 7.24426392605560812004831114944, 8.771564764037043188316300595372, 10.09643764756373628572642491148, 10.55927669873970150735252321856, 11.67336574669003454752376285036

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