Properties

Label 2-162-3.2-c2-0-5
Degree $2$
Conductor $162$
Sign $i$
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  − 1.41i·2-s − 2.00·4-s − 1.55i·5-s + 12.3·7-s + 2.82i·8-s − 2.19·10-s − 14.6i·11-s − 10.8·13-s − 17.5i·14-s + 4.00·16-s − 28.9i·17-s + 3.60·19-s + 3.10i·20-s − 20.7·22-s + 14.6i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.500·4-s − 0.310i·5-s + 1.77·7-s + 0.353i·8-s − 0.219·10-s − 1.33i·11-s − 0.831·13-s − 1.25i·14-s + 0.250·16-s − 1.70i·17-s + 0.189·19-s + 0.155i·20-s − 0.944·22-s + 0.638i·23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $i$
Analytic conductor: \(4.41418\)
Root analytic conductor: \(2.10099\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :1),\ i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.09891 - 1.09891i\)
\(L(\frac12)\) \(\approx\) \(1.09891 - 1.09891i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 \)
good5 \( 1 + 1.55iT - 25T^{2} \)
7 \( 1 - 12.3T + 49T^{2} \)
11 \( 1 + 14.6iT - 121T^{2} \)
13 \( 1 + 10.8T + 169T^{2} \)
17 \( 1 + 28.9iT - 289T^{2} \)
19 \( 1 - 3.60T + 361T^{2} \)
23 \( 1 - 14.6iT - 529T^{2} \)
29 \( 1 - 28.1iT - 841T^{2} \)
31 \( 1 - 8T + 961T^{2} \)
37 \( 1 - 22.5T + 1.36e3T^{2} \)
41 \( 1 - 25.1iT - 1.68e3T^{2} \)
43 \( 1 + 53.1T + 1.84e3T^{2} \)
47 \( 1 + 16.9iT - 2.20e3T^{2} \)
53 \( 1 - 84.5iT - 2.80e3T^{2} \)
59 \( 1 - 91.0iT - 3.48e3T^{2} \)
61 \( 1 + 13T + 3.72e3T^{2} \)
67 \( 1 + 41.1T + 4.48e3T^{2} \)
71 \( 1 - 16.3iT - 5.04e3T^{2} \)
73 \( 1 - 71.5T + 5.32e3T^{2} \)
79 \( 1 + 46.7T + 6.24e3T^{2} \)
83 \( 1 - 15.3iT - 6.88e3T^{2} \)
89 \( 1 + 78.9iT - 7.92e3T^{2} \)
97 \( 1 - 91.1T + 9.40e3T^{2} \)
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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

−12.01830112138569347982127989533, −11.47367488945750550745271304388, −10.65445508643269214074624345903, −9.280163248393608496049934331649, −8.409774595413748179119831159457, −7.38556907458689586132010136765, −5.37867930653519245751032729566, −4.65771532530577310090220744271, −2.86487576909049947862593703507, −1.13041756349053652121415970419, 1.94274724425576045271595004294, 4.31801430535720735154613493551, 5.13477280980763541948708370669, 6.64288782676863942523476843447, 7.73411137293345524284399354022, 8.411301001047979272813089975474, 9.846473497852346163986420320802, 10.79787237351582273744794597635, 11.97113825818170461211529781036, 12.89165719914046000139989002731

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