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Properties

Label 2-162-9.2-c8-0-29
Degree $2$
Conductor $162$
Sign $-0.939 + 0.342i$
Analytic cond. $65.9953$
Root an. cond. $8.12375$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (9.79 + 5.65i)2-s + (63.9 + 110. i)4-s + (−202. + 116. i)5-s + (1.76e3 − 3.05e3i)7-s + 1.44e3i·8-s − 2.64e3·10-s + (−1.74e4 − 1.00e4i)11-s + (2.09e4 + 3.62e4i)13-s + (3.46e4 − 1.99e4i)14-s + (−8.19e3 + 1.41e4i)16-s − 9.47e4i·17-s − 3.63e4·19-s + (−2.58e4 − 1.49e4i)20-s + (−1.14e5 − 1.97e5i)22-s + (−3.58e5 + 2.06e5i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.323 + 0.186i)5-s + (0.735 − 1.27i)7-s + 0.353i·8-s − 0.264·10-s + (−1.19 − 0.689i)11-s + (0.732 + 1.26i)13-s + (0.900 − 0.520i)14-s + (−0.125 + 0.216i)16-s − 1.13i·17-s − 0.278·19-s + (−0.161 − 0.0933i)20-s + (−0.487 − 0.843i)22-s + (−1.28 + 0.739i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $-0.939 + 0.342i$
Analytic conductor: \(65.9953\)
Root analytic conductor: \(8.12375\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :4),\ -0.939 + 0.342i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.04765418853\)
\(L(\frac12)\) \(\approx\) \(0.04765418853\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-9.79 - 5.65i)T \)
3 \( 1 \)
good5 \( 1 + (202. - 116. i)T + (1.95e5 - 3.38e5i)T^{2} \)
7 \( 1 + (-1.76e3 + 3.05e3i)T + (-2.88e6 - 4.99e6i)T^{2} \)
11 \( 1 + (1.74e4 + 1.00e4i)T + (1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 + (-2.09e4 - 3.62e4i)T + (-4.07e8 + 7.06e8i)T^{2} \)
17 \( 1 + 9.47e4iT - 6.97e9T^{2} \)
19 \( 1 + 3.63e4T + 1.69e10T^{2} \)
23 \( 1 + (3.58e5 - 2.06e5i)T + (3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 + (-2.33e5 - 1.34e5i)T + (2.50e11 + 4.33e11i)T^{2} \)
31 \( 1 + (-2.35e5 - 4.08e5i)T + (-4.26e11 + 7.38e11i)T^{2} \)
37 \( 1 + 3.00e6T + 3.51e12T^{2} \)
41 \( 1 + (-1.48e6 + 8.57e5i)T + (3.99e12 - 6.91e12i)T^{2} \)
43 \( 1 + (1.81e6 - 3.13e6i)T + (-5.84e12 - 1.01e13i)T^{2} \)
47 \( 1 + (5.20e6 + 3.00e6i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + 1.02e7iT - 6.22e13T^{2} \)
59 \( 1 + (2.32e6 - 1.34e6i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (-2.72e6 + 4.71e6i)T + (-9.58e13 - 1.66e14i)T^{2} \)
67 \( 1 + (-3.06e6 - 5.30e6i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 - 2.11e7iT - 6.45e14T^{2} \)
73 \( 1 + 4.90e7T + 8.06e14T^{2} \)
79 \( 1 + (4.17e6 - 7.23e6i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 + (4.45e7 + 2.56e7i)T + (1.12e15 + 1.95e15i)T^{2} \)
89 \( 1 + 1.07e8iT - 3.93e15T^{2} \)
97 \( 1 + (1.02e7 - 1.76e7i)T + (-3.91e15 - 6.78e15i)T^{2} \)
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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

−11.18549761833676476016506841262, −10.10847765662922475138908097728, −8.513656983086907250675066761721, −7.61144252629528010938643465957, −6.77920982653387373527563812178, −5.36977384135486571567675055666, −4.30723050685683662143424389026, −3.32016597667790960233876708569, −1.64184927315158007312631400048, −0.008486098359518228685904085067, 1.76667096195626605720258105394, 2.73679093895844601324093149332, 4.21833726137295965009447905953, 5.33108743474238493770738208658, 6.11155617419926936640205377094, 7.971371501744652543209132209565, 8.456942602101387577502328478295, 10.13565975327560591291716253172, 10.82299129864618253570101195994, 12.12838395587736057996984966655

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