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Properties

Label 2-162-9.5-c2-0-2
Degree $2$
Conductor $162$
Sign $0.766 - 0.642i$
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 + (7.34 + 4.24i)5-s + (−2.5 − 4.33i)7-s + 2.82i·8-s − 12·10-s + (7.34 − 4.24i)11-s + (0.5 − 0.866i)13-s + (6.12 + 3.53i)14-s + (−2.00 − 3.46i)16-s + 25.4i·17-s + 29·19-s + (14.6 − 8.48i)20-s + (−6 + 10.3i)22-s + (7.34 + 4.24i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (1.46 + 0.848i)5-s + (−0.357 − 0.618i)7-s + 0.353i·8-s − 1.20·10-s + (0.668 − 0.385i)11-s + (0.0384 − 0.0666i)13-s + (0.437 + 0.252i)14-s + (−0.125 − 0.216i)16-s + 1.49i·17-s + 1.52·19-s + (0.734 − 0.424i)20-s + (−0.272 + 0.472i)22-s + (0.319 + 0.184i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.766 - 0.642i$
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: \((2,\ 162,\ (\ :1),\ 0.766 - 0.642i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.30751 + 0.475897i\)
\(L(\frac12)\) \(\approx\) \(1.30751 + 0.475897i\)
\(L(2)\) 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 \( 1 + (-7.34 - 4.24i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (2.5 + 4.33i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-7.34 + 4.24i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 25.4iT - 289T^{2} \)
19 \( 1 - 29T + 361T^{2} \)
23 \( 1 + (-7.34 - 4.24i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (14.6 - 8.48i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-5 + 8.66i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 25T + 1.36e3T^{2} \)
41 \( 1 + (14.6 + 8.48i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (7 + 12.1i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-7.34 + 4.24i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 50.9iT - 2.80e3T^{2} \)
59 \( 1 + (80.8 + 46.6i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (11.5 + 19.9i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-9.5 + 16.4i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 101. iT - 5.04e3T^{2} \)
73 \( 1 + 97T + 5.32e3T^{2} \)
79 \( 1 + (38.5 + 66.6i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (102. - 59.3i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 76.3iT - 7.92e3T^{2} \)
97 \( 1 + (-24.5 - 42.4i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−13.00428038137347713992370891773, −11.40861855773973059170937881736, −10.39089080409799139168734407912, −9.830923665424641463362971322449, −8.813096840195227283944095371187, −7.31402114981213964124098180618, −6.42787180637193247668204617487, −5.56861421852791701842672904333, −3.37717430426346081193977805578, −1.59505310387997266944460933672, 1.33133852403946433233456482866, 2.79217742867388587121974580904, 4.92877923061377971168191209855, 6.01945073639271631169739804103, 7.32069222616890723249216268905, 9.014271242020913458951775813444, 9.305200659587961045607635419072, 10.14500863390879408455770720106, 11.65774838736681014169060708207, 12.37890640300314117319800302631

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