Properties

Label 2-162-27.22-c1-0-1
Degree $2$
Conductor $162$
Sign $0.984 + 0.177i$
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  + (−0.766 + 0.642i)2-s + (0.173 − 0.984i)4-s + (1.96 − 0.714i)5-s + (−0.696 − 3.95i)7-s + (0.500 + 0.866i)8-s + (−1.04 + 1.80i)10-s + (0.199 + 0.0726i)11-s + (3.98 + 3.34i)13-s + (3.07 + 2.57i)14-s + (−0.939 − 0.342i)16-s + (1.89 − 3.27i)17-s + (0.636 + 1.10i)19-s + (−0.362 − 2.05i)20-s + (−0.199 + 0.0726i)22-s + (−0.144 + 0.816i)23-s + ⋯
L(s)  = 1  + (−0.541 + 0.454i)2-s + (0.0868 − 0.492i)4-s + (0.877 − 0.319i)5-s + (−0.263 − 1.49i)7-s + (0.176 + 0.306i)8-s + (−0.330 + 0.571i)10-s + (0.0601 + 0.0219i)11-s + (1.10 + 0.926i)13-s + (0.821 + 0.689i)14-s + (−0.234 − 0.0855i)16-s + (0.459 − 0.795i)17-s + (0.146 + 0.252i)19-s + (−0.0810 − 0.459i)20-s + (−0.0425 + 0.0154i)22-s + (−0.0300 + 0.170i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.984 + 0.177i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :1/2),\ 0.984 + 0.177i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.991526 - 0.0887991i\)
\(L(\frac12)\) \(\approx\) \(0.991526 - 0.0887991i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 - 0.642i)T \)
3 \( 1 \)
good5 \( 1 + (-1.96 + 0.714i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (0.696 + 3.95i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (-0.199 - 0.0726i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (-3.98 - 3.34i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-1.89 + 3.27i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.636 - 1.10i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.144 - 0.816i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (7.05 - 5.92i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-0.614 + 3.48i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (1.77 - 3.07i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.09 + 1.76i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (6.48 + 2.35i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.221 - 1.25i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 + (8.04 - 2.92i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-0.492 - 2.79i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-7.47 - 6.27i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (5.86 - 10.1i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.375 + 0.649i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.81 - 1.52i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-8.73 + 7.32i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (-2.69 - 4.67i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-11.8 - 4.29i)T + (74.3 + 62.3i)T^{2} \)
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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

−13.32941605669454957754362338646, −11.63082452338115259428441745595, −10.56651170494777889007414713447, −9.727488713545481864368058432813, −8.880382319916555578585072202866, −7.49208852324356082102916366031, −6.62991842510868440245538611549, −5.41438245013610879922457912194, −3.84762208731890856364054872841, −1.40546970735566747806752543355, 2.02873927601914779256361266137, 3.34671227122929663997204631129, 5.56646102629885148838068251510, 6.30609659532932238380644180315, 8.046500745568108665553809779317, 8.939045260025894923860278058013, 9.852167193810737752539959113651, 10.76764188383350232531469575485, 11.85751387571631655462196766872, 12.79729605875323130299600197591

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