↑

Properties

Label 2-2160-4.3-c2-0-29
Degree $2$
Conductor $2160$
Sign $0.866 + 0.5i$
Analytic cond. $58.8557$
Root an. cond. $7.67174$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.23·5-s + 2.77i·7-s + 9.66i·11-s − 24.3·13-s − 13.0·17-s − 3.18i·19-s − 35.0i·23-s + 5.00·25-s + 1.74·29-s + 10.7i·31-s − 6.20i·35-s + 48.0·37-s − 61.9·41-s − 79.8i·43-s + 68.0i·47-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.396i·7-s + 0.878i·11-s − 1.87·13-s − 0.766·17-s − 0.167i·19-s − 1.52i·23-s + 0.200·25-s + 0.0600·29-s + 0.348i·31-s − 0.177i·35-s + 1.29·37-s − 1.51·41-s − 1.85i·43-s + 1.44i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $0.866 + 0.5i$
Analytic conductor: \(58.8557\)
Root analytic conductor: \(7.67174\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :1),\ 0.866 + 0.5i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.126778103\)
\(L(\frac12)\) \(\approx\) \(1.126778103\)
\(L(2)\) 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 \( 1 \)
5 \( 1 + 2.23T \)
good7 \( 1 - 2.77iT - 49T^{2} \)
11 \( 1 - 9.66iT - 121T^{2} \)
13 \( 1 + 24.3T + 169T^{2} \)
17 \( 1 + 13.0T + 289T^{2} \)
19 \( 1 + 3.18iT - 361T^{2} \)
23 \( 1 + 35.0iT - 529T^{2} \)
29 \( 1 - 1.74T + 841T^{2} \)
31 \( 1 - 10.7iT - 961T^{2} \)
37 \( 1 - 48.0T + 1.36e3T^{2} \)
41 \( 1 + 61.9T + 1.68e3T^{2} \)
43 \( 1 + 79.8iT - 1.84e3T^{2} \)
47 \( 1 - 68.0iT - 2.20e3T^{2} \)
53 \( 1 - 90.7T + 2.80e3T^{2} \)
59 \( 1 - 13.1iT - 3.48e3T^{2} \)
61 \( 1 - 48.8T + 3.72e3T^{2} \)
67 \( 1 + 69.2iT - 4.48e3T^{2} \)
71 \( 1 - 31.5iT - 5.04e3T^{2} \)
73 \( 1 - 77.6T + 5.32e3T^{2} \)
79 \( 1 - 139. iT - 6.24e3T^{2} \)
83 \( 1 - 13.5iT - 6.88e3T^{2} \)
89 \( 1 + 129.T + 7.92e3T^{2} \)
97 \( 1 - 43.2T + 9.40e3T^{2} \)
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

−8.795922043588671655241304287815, −8.089874903190098409894852036885, −7.08076038092580626779392578902, −6.82946141357884282573469505196, −5.51094702529529026917457598122, −4.72545621348663238056092707395, −4.14251327517314800293652424781, −2.69389391515797974127538165931, −2.16173144054360842215411121258, −0.41690317864693972100751639104, 0.68654371311632483583347922561, 2.14825175632873708562793214482, 3.15495541490717399903320651705, 4.07462001426771727564995578428, 4.92732943389691838692535070447, 5.72686819447479669418433723276, 6.79832521714080685531687418438, 7.41994843551402420229276561282, 8.088057255169682538692062182907, 8.947483020038391356159628399601

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