Properties

Label 2-162-3.2-c6-0-13
Degree $2$
Conductor $162$
Sign $i$
Analytic cond. $37.2687$
Root an. cond. $6.10481$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.65i·2-s − 32.0·4-s + 24.8i·5-s − 6.19·7-s + 181. i·8-s + 140.·10-s − 130. i·11-s − 922.·13-s + 35.0i·14-s + 1.02e3·16-s + 3.38e3i·17-s + 5.40e3·19-s − 794. i·20-s − 740.·22-s − 2.36e3i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.500·4-s + 0.198i·5-s − 0.0180·7-s + 0.353i·8-s + 0.140·10-s − 0.0983i·11-s − 0.419·13-s + 0.0127i·14-s + 0.250·16-s + 0.688i·17-s + 0.787·19-s − 0.0992i·20-s − 0.0695·22-s − 0.194i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $i$
Analytic conductor: \(37.2687\)
Root analytic conductor: \(6.10481\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :3),\ i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.666957347\)
\(L(\frac12)\) \(\approx\) \(1.666957347\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 5.65iT \)
3 \( 1 \)
good5 \( 1 - 24.8iT - 1.56e4T^{2} \)
7 \( 1 + 6.19T + 1.17e5T^{2} \)
11 \( 1 + 130. iT - 1.77e6T^{2} \)
13 \( 1 + 922.T + 4.82e6T^{2} \)
17 \( 1 - 3.38e3iT - 2.41e7T^{2} \)
19 \( 1 - 5.40e3T + 4.70e7T^{2} \)
23 \( 1 + 2.36e3iT - 1.48e8T^{2} \)
29 \( 1 + 3.49e4iT - 5.94e8T^{2} \)
31 \( 1 - 4.66e3T + 8.87e8T^{2} \)
37 \( 1 - 6.91e3T + 2.56e9T^{2} \)
41 \( 1 + 5.38e4iT - 4.75e9T^{2} \)
43 \( 1 - 1.23e5T + 6.32e9T^{2} \)
47 \( 1 + 9.60e4iT - 1.07e10T^{2} \)
53 \( 1 + 1.32e5iT - 2.21e10T^{2} \)
59 \( 1 + 2.90e4iT - 4.21e10T^{2} \)
61 \( 1 - 3.20e5T + 5.15e10T^{2} \)
67 \( 1 + 5.29e5T + 9.04e10T^{2} \)
71 \( 1 + 6.28e5iT - 1.28e11T^{2} \)
73 \( 1 + 6.87e4T + 1.51e11T^{2} \)
79 \( 1 - 4.71e5T + 2.43e11T^{2} \)
83 \( 1 + 3.35e4iT - 3.26e11T^{2} \)
89 \( 1 + 1.01e6iT - 4.96e11T^{2} \)
97 \( 1 - 4.18e5T + 8.32e11T^{2} \)
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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

−11.50318721116501458078081881933, −10.52959532295513050359998844929, −9.662541171565286228910613855795, −8.564081670731083971015383943257, −7.40765341690456207880144272247, −6.01337324632194373119299402676, −4.70088089819609813924858913189, −3.41748306930860739481483671933, −2.15065574576836204074426493832, −0.60825224092598202943167468638, 1.02765213242165244982656942075, 2.97299123776464371834841582617, 4.55643415361811843409152634271, 5.51357133450215452671542029177, 6.83792085026262081554111260614, 7.69911966681077427095742372652, 8.909182436333743728544929509819, 9.721681722110058819379387556930, 10.97009340293895737513523714509, 12.17273286561426335234510804519

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