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Properties

Label 2-162-9.5-c2-0-5
Degree $2$
Conductor $162$
Sign $-0.0871 + 0.996i$
Analytic cond. $4.41418$
Root an. cond. $2.10099$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 + 0.707i)2-s + (0.999 − 1.73i)4-s + (1.34 + 0.776i)5-s + (−6.19 − 10.7i)7-s + 2.82i·8-s − 2.19·10-s + (−12.7 + 7.34i)11-s + (5.40 − 9.35i)13-s + (15.1 + 8.76i)14-s + (−2.00 − 3.46i)16-s − 28.9i·17-s + 3.60·19-s + (2.68 − 1.55i)20-s + (10.3 − 18i)22-s + (−12.7 − 7.34i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.268 + 0.155i)5-s + (−0.885 − 1.53i)7-s + 0.353i·8-s − 0.219·10-s + (−1.15 + 0.668i)11-s + (0.415 − 0.719i)13-s + (1.08 + 0.625i)14-s + (−0.125 − 0.216i)16-s − 1.70i·17-s + 0.189·19-s + (0.134 − 0.0776i)20-s + (0.472 − 0.818i)22-s + (−0.553 − 0.319i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0871 + 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0871 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $-0.0871 + 0.996i$
Analytic conductor: \(4.41418\)
Root analytic conductor: \(2.10099\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :1),\ -0.0871 + 0.996i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.466044 - 0.508598i\)
\(L(\frac12)\) \(\approx\) \(0.466044 - 0.508598i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 - 0.707i)T \)
3 \( 1 \)
good5 \( 1 + (-1.34 - 0.776i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (6.19 + 10.7i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (12.7 - 7.34i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-5.40 + 9.35i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 28.9iT - 289T^{2} \)
19 \( 1 - 3.60T + 361T^{2} \)
23 \( 1 + (12.7 + 7.34i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-24.3 + 14.0i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 22.5T + 1.36e3T^{2} \)
41 \( 1 + (21.7 + 12.5i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-26.5 - 46.0i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (14.6 - 8.48i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 84.5iT - 2.80e3T^{2} \)
59 \( 1 + (78.8 + 45.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-6.5 - 11.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-20.5 + 35.6i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 16.3iT - 5.04e3T^{2} \)
73 \( 1 - 71.5T + 5.32e3T^{2} \)
79 \( 1 + (-23.3 - 40.4i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-13.2 + 7.65i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 78.9iT - 7.92e3T^{2} \)
97 \( 1 + (45.5 + 78.9i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.48318561099264874360916114280, −10.96218627336715322004755520396, −10.14165658307706052182948834364, −9.621820573881083746655788553888, −7.951241442757592247865521532467, −7.24953859199507785794763403346, −6.16988317888333559959355456395, −4.64428810072702964938002112695, −2.86360404950262065881847506330, −0.50119531386977477575680577539, 2.08589254410539527013732739462, 3.43765566328830062737882266674, 5.53999541989370817658652714399, 6.38464612737493382651327751429, 8.102897812426209967054027357792, 8.854846691773620558258674933798, 9.773563792235685842309601362990, 10.80309290337147781634214210439, 11.90629914194047985036683521179, 12.76076881780952954369077958668

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