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Properties

Label 2-162-9.4-c1-0-0
Degree $2$
Conductor $162$
Sign $0.766 - 0.642i$
Analytic cond. $1.29357$
Root an. cond. $1.13735$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−1.5 + 2.59i)5-s + (2 + 3.46i)7-s + 0.999·8-s + 3·10-s + (0.5 − 0.866i)13-s + (1.99 − 3.46i)14-s + (−0.5 − 0.866i)16-s + 3·17-s − 4·19-s + (−1.50 − 2.59i)20-s + (−2 − 3.46i)25-s − 0.999·26-s − 3.99·28-s + (4.5 + 7.79i)29-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.670 + 1.16i)5-s + (0.755 + 1.30i)7-s + 0.353·8-s + 0.948·10-s + (0.138 − 0.240i)13-s + (0.534 − 0.925i)14-s + (−0.125 − 0.216i)16-s + 0.727·17-s − 0.917·19-s + (−0.335 − 0.580i)20-s + (−0.400 − 0.692i)25-s − 0.196·26-s − 0.755·28-s + (0.835 + 1.44i)29-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :1/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.816019 + 0.297006i\)
\(L(\frac12)\) \(\approx\) \(0.816019 + 0.297006i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
good5 \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2 - 3.46i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.5 - 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 + (-8 - 13.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.56092277731698895988134750242, −11.83773769543142733528718305468, −11.03558759433353539472538089512, −10.21551095504308587263727186398, −8.791583683002349525075838870308, −8.037100939293410167216438338055, −6.78184996924851581153455801714, −5.26256129327498560513843139936, −3.56603547037205579114074833461, −2.30857988801074308456157808772, 1.04188007843521944679127450189, 4.14601703291394264621390941545, 4.88409631804608462809178866932, 6.52059292934436182742534778741, 7.902141144098445941830875261832, 8.225980947945563300448616123922, 9.579885372707630361047891405271, 10.68085225261900373649094938870, 11.72820327580494862847660403966, 12.85359215630589938581548332257

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