Properties

Label 2-162-3.2-c6-0-11
Degree $2$
Conductor $162$
Sign $1$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5.65i·2-s − 32.0·4-s − 10.8i·5-s − 644.·7-s − 181. i·8-s + 61.3·10-s + 1.22e3i·11-s − 2.32e3·13-s − 3.64e3i·14-s + 1.02e3·16-s + 3.38e3i·17-s + 6.23e3·19-s + 347. i·20-s − 6.91e3·22-s − 1.43e4i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.500·4-s − 0.0867i·5-s − 1.87·7-s − 0.353i·8-s + 0.0613·10-s + 0.918i·11-s − 1.05·13-s − 1.32i·14-s + 0.250·16-s + 0.688i·17-s + 0.908·19-s + 0.0433i·20-s − 0.649·22-s − 1.18i·23-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.9448493447\)
\(L(\frac12)\) \(\approx\) \(0.9448493447\)
\(L(4)\) 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 \( 1 + 10.8iT - 1.56e4T^{2} \)
7 \( 1 + 644.T + 1.17e5T^{2} \)
11 \( 1 - 1.22e3iT - 1.77e6T^{2} \)
13 \( 1 + 2.32e3T + 4.82e6T^{2} \)
17 \( 1 - 3.38e3iT - 2.41e7T^{2} \)
19 \( 1 - 6.23e3T + 4.70e7T^{2} \)
23 \( 1 + 1.43e4iT - 1.48e8T^{2} \)
29 \( 1 + 1.35e4iT - 5.94e8T^{2} \)
31 \( 1 - 8.78e3T + 8.87e8T^{2} \)
37 \( 1 + 2.78e4T + 2.56e9T^{2} \)
41 \( 1 + 5.53e4iT - 4.75e9T^{2} \)
43 \( 1 - 5.39e4T + 6.32e9T^{2} \)
47 \( 1 + 1.76e5iT - 1.07e10T^{2} \)
53 \( 1 + 8.39e4iT - 2.21e10T^{2} \)
59 \( 1 - 3.53e5iT - 4.21e10T^{2} \)
61 \( 1 - 1.66e5T + 5.15e10T^{2} \)
67 \( 1 + 9.04e4T + 9.04e10T^{2} \)
71 \( 1 - 4.29e5iT - 1.28e11T^{2} \)
73 \( 1 - 2.74e5T + 1.51e11T^{2} \)
79 \( 1 - 3.26e5T + 2.43e11T^{2} \)
83 \( 1 - 9.91e5iT - 3.26e11T^{2} \)
89 \( 1 + 5.38e5iT - 4.96e11T^{2} \)
97 \( 1 + 8.59e4T + 8.32e11T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.24242218530905639459350170124, −10.27571577580298607418341145001, −9.739591505605735714306850925896, −8.713166314666710937837367812142, −7.23162297344091754803576810066, −6.63980009074166621225626813494, −5.39276510067912347676974321659, −4.03523365186633362800551745369, −2.64508492337470199149923236511, −0.40627060978729479606589682021, 0.77838272011418744674074255293, 2.81044409512412608638054784830, 3.42900631679319475030846540848, 5.14068665413548820270943167932, 6.39288902158801095801939934963, 7.54157306614644361548613244455, 9.198969058820721316576615050332, 9.634119594099514460298703650888, 10.70785582750317136550067042353, 11.84225904714434978923884822838

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