Properties

Label 2-162-9.7-c3-0-9
Degree $2$
Conductor $162$
Sign $-0.642 + 0.766i$
Analytic cond. $9.55830$
Root an. cond. $3.09165$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)2-s + (−1.99 − 3.46i)4-s + (0.401 + 0.696i)5-s + (1.19 − 2.07i)7-s − 7.99·8-s + 1.60·10-s + (29.7 − 51.5i)11-s + (−38.8 − 67.3i)13-s + (−2.39 − 4.14i)14-s + (−8 + 13.8i)16-s − 2.84·17-s − 118.·19-s + (1.60 − 2.78i)20-s + (−59.5 − 103. i)22-s + (55.3 + 95.9i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.0359 + 0.0622i)5-s + (0.0645 − 0.111i)7-s − 0.353·8-s + 0.0508·10-s + (0.816 − 1.41i)11-s + (−0.829 − 1.43i)13-s + (−0.0456 − 0.0791i)14-s + (−0.125 + 0.216i)16-s − 0.0405·17-s − 1.42·19-s + (0.0179 − 0.0311i)20-s + (−0.577 − 0.999i)22-s + (0.502 + 0.869i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $-0.642 + 0.766i$
Analytic conductor: \(9.55830\)
Root analytic conductor: \(3.09165\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :3/2),\ -0.642 + 0.766i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.698853 - 1.49869i\)
\(L(\frac12)\) \(\approx\) \(0.698853 - 1.49869i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + 1.73i)T \)
3 \( 1 \)
good5 \( 1 + (-0.401 - 0.696i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (-1.19 + 2.07i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-29.7 + 51.5i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (38.8 + 67.3i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 2.84T + 4.91e3T^{2} \)
19 \( 1 + 118.T + 6.85e3T^{2} \)
23 \( 1 + (-55.3 - 95.9i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-62.8 + 108. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (31.5 + 54.6i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 227.T + 5.06e4T^{2} \)
41 \( 1 + (-162. - 281. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-136. + 236. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (2.46 - 4.27i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 598.T + 1.48e5T^{2} \)
59 \( 1 + (335. + 580. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (232. - 402. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-384. - 666. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 611.T + 3.57e5T^{2} \)
73 \( 1 - 923.T + 3.89e5T^{2} \)
79 \( 1 + (19.9 - 34.4i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-221. + 384. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 78.4T + 7.04e5T^{2} \)
97 \( 1 + (-45.5 + 78.8i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.07381464315589714086867510273, −11.01533029501834914751467529995, −10.30640346482494769152967393396, −9.051021056674815341533729606896, −8.026126270263768653965562311445, −6.45398174561366671329717195593, −5.37497408307283654403195497968, −3.93231842492089854172311952094, −2.66798676908221348239208489808, −0.69839807334266746232937105452, 2.08149992295496464539812355347, 4.12272871773412943838636634483, 4.96456807204311572963434566352, 6.65619446453964322406335967405, 7.12960693755106879918702996602, 8.726353430105103288298105878155, 9.444387806220729893309477083920, 10.79691735827895805721701113076, 12.19317362840240251547709232872, 12.56647036603890525523623183489

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