Properties

Label 2-6e3-3.2-c2-0-5
Degree $2$
Conductor $216$
Sign $i$
Analytic cond. $5.88557$
Root an. cond. $2.42602$
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  − 2.82i·5-s − 3·7-s − 19.7i·11-s + 7·13-s − 14.1i·17-s − 19·19-s − 2.82i·23-s + 17·25-s − 50.9i·29-s − 10·31-s + 8.48i·35-s + 63·37-s + 50.9i·41-s − 50·43-s − 42.4i·47-s + ⋯
L(s)  = 1  − 0.565i·5-s − 0.428·7-s − 1.79i·11-s + 0.538·13-s − 0.831i·17-s − 19-s − 0.122i·23-s + 0.680·25-s − 1.75i·29-s − 0.322·31-s + 0.242i·35-s + 1.70·37-s + 1.24i·41-s − 1.16·43-s − 0.902i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.892200 - 0.892200i\)
\(L(\frac12)\) \(\approx\) \(0.892200 - 0.892200i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + 2.82iT - 25T^{2} \)
7 \( 1 + 3T + 49T^{2} \)
11 \( 1 + 19.7iT - 121T^{2} \)
13 \( 1 - 7T + 169T^{2} \)
17 \( 1 + 14.1iT - 289T^{2} \)
19 \( 1 + 19T + 361T^{2} \)
23 \( 1 + 2.82iT - 529T^{2} \)
29 \( 1 + 50.9iT - 841T^{2} \)
31 \( 1 + 10T + 961T^{2} \)
37 \( 1 - 63T + 1.36e3T^{2} \)
41 \( 1 - 50.9iT - 1.68e3T^{2} \)
43 \( 1 + 50T + 1.84e3T^{2} \)
47 \( 1 + 42.4iT - 2.20e3T^{2} \)
53 \( 1 - 73.5iT - 2.80e3T^{2} \)
59 \( 1 - 98.9iT - 3.48e3T^{2} \)
61 \( 1 - 79T + 3.72e3T^{2} \)
67 \( 1 - 77T + 4.48e3T^{2} \)
71 \( 1 - 79.1iT - 5.04e3T^{2} \)
73 \( 1 + 17T + 5.32e3T^{2} \)
79 \( 1 + 11T + 6.24e3T^{2} \)
83 \( 1 + 39.5iT - 6.88e3T^{2} \)
89 \( 1 - 42.4iT - 7.92e3T^{2} \)
97 \( 1 + 97T + 9.40e3T^{2} \)
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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

−11.70977860610617227098567862651, −11.04506956382298680623223183801, −9.825943646897817915613913547795, −8.774299524182894108602763993439, −8.101326986853750701934163540746, −6.53007635444527891597178741168, −5.67992556246414970759367312034, −4.24801590395536063941728662376, −2.88492772039715598952188767432, −0.71148437845598326309032742967, 1.98666229873596286392538499627, 3.57282744992988254813882842688, 4.86084522784574180344487689848, 6.40110719362524553800153061383, 7.09492117274099749645533492537, 8.359929886661361973947746881358, 9.539435407952949826363109546005, 10.39482482914812763809836030204, 11.22638498827461634904094621148, 12.69305762787801526711545289047

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