Properties

Label 2-220-220.43-c2-0-39
Degree $2$
Conductor $220$
Sign $0.586 - 0.809i$
Analytic cond. $5.99456$
Root an. cond. $2.44838$
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  + (−1.49 + 1.33i)2-s + (−3.87 + 3.87i)3-s + (0.444 − 3.97i)4-s + (3.64 + 3.41i)5-s + (0.609 − 10.9i)6-s + (5.44 − 5.44i)7-s + (4.63 + 6.51i)8-s − 21.0i·9-s + (−9.99 − 0.233i)10-s + (7.73 − 7.82i)11-s + (13.6 + 17.1i)12-s + (14.7 − 14.7i)13-s + (−0.856 + 15.3i)14-s + (−27.3 + 0.885i)15-s + (−15.6 − 3.53i)16-s + (5.69 + 5.69i)17-s + ⋯
L(s)  = 1  + (−0.745 + 0.666i)2-s + (−1.29 + 1.29i)3-s + (0.111 − 0.993i)4-s + (0.729 + 0.683i)5-s + (0.101 − 1.82i)6-s + (0.778 − 0.778i)7-s + (0.579 + 0.814i)8-s − 2.33i·9-s + (−0.999 − 0.0233i)10-s + (0.703 − 0.711i)11-s + (1.14 + 1.42i)12-s + (1.13 − 1.13i)13-s + (−0.0611 + 1.09i)14-s + (−1.82 + 0.0590i)15-s + (−0.975 − 0.220i)16-s + (0.335 + 0.335i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 - 0.809i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.586 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.586 - 0.809i$
Analytic conductor: \(5.99456\)
Root analytic conductor: \(2.44838\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1),\ 0.586 - 0.809i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.792919 + 0.404756i\)
\(L(\frac12)\) \(\approx\) \(0.792919 + 0.404756i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.49 - 1.33i)T \)
5 \( 1 + (-3.64 - 3.41i)T \)
11 \( 1 + (-7.73 + 7.82i)T \)
good3 \( 1 + (3.87 - 3.87i)T - 9iT^{2} \)
7 \( 1 + (-5.44 + 5.44i)T - 49iT^{2} \)
13 \( 1 + (-14.7 + 14.7i)T - 169iT^{2} \)
17 \( 1 + (-5.69 - 5.69i)T + 289iT^{2} \)
19 \( 1 + 21.1iT - 361T^{2} \)
23 \( 1 + (7.64 - 7.64i)T - 529iT^{2} \)
29 \( 1 + 2.11T + 841T^{2} \)
31 \( 1 + 23.3iT - 961T^{2} \)
37 \( 1 + (-7.12 + 7.12i)T - 1.36e3iT^{2} \)
41 \( 1 + 13.4iT - 1.68e3T^{2} \)
43 \( 1 + (18.4 + 18.4i)T + 1.84e3iT^{2} \)
47 \( 1 + (25.7 + 25.7i)T + 2.20e3iT^{2} \)
53 \( 1 + (10.8 + 10.8i)T + 2.80e3iT^{2} \)
59 \( 1 - 84.0T + 3.48e3T^{2} \)
61 \( 1 + 87.3iT - 3.72e3T^{2} \)
67 \( 1 + (-55.9 - 55.9i)T + 4.48e3iT^{2} \)
71 \( 1 - 24.0iT - 5.04e3T^{2} \)
73 \( 1 + (69.0 - 69.0i)T - 5.32e3iT^{2} \)
79 \( 1 - 47.8iT - 6.24e3T^{2} \)
83 \( 1 + (-11.0 - 11.0i)T + 6.88e3iT^{2} \)
89 \( 1 - 90.4iT - 7.92e3T^{2} \)
97 \( 1 + (88.2 - 88.2i)T - 9.40e3iT^{2} \)
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

−11.33582164289265109725122307635, −11.07085475410711347237272275245, −10.31426052095084833183736170976, −9.555169148725448761473598565592, −8.371003238515871584375821639528, −6.85603534538499487708329220563, −5.94434139935594884216300024296, −5.25641108020763612202471494900, −3.80586309434548941209182496569, −0.892092232989394047878275942689, 1.36095736882519359820680363037, 1.87411918470676600138233859291, 4.59839933582625659269249073081, 5.90136789284652523176379876822, 6.77992707693334667036064554460, 8.047422303778036267286800781502, 8.900930662605223989052789140794, 10.09859475104597464466237018139, 11.30428689959934926944202566742, 11.90376207433208026256494171715

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