↑

Properties

Label 2-220-4.3-c2-0-5
Degree $2$
Conductor $220$
Sign $-0.638 + 0.769i$
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  + (−0.679 + 1.88i)2-s + 5.18i·3-s + (−3.07 − 2.55i)4-s + 2.23·5-s + (−9.75 − 3.52i)6-s + 11.9i·7-s + (6.89 − 4.05i)8-s − 17.8·9-s + (−1.51 + 4.20i)10-s − 3.31i·11-s + (13.2 − 15.9i)12-s − 14.0·13-s + (−22.5 − 8.13i)14-s + 11.5i·15-s + (2.94 + 15.7i)16-s + 20.7·17-s + ⋯
L(s)  = 1  + (−0.339 + 0.940i)2-s + 1.72i·3-s + (−0.769 − 0.638i)4-s + 0.447·5-s + (−1.62 − 0.587i)6-s + 1.71i·7-s + (0.862 − 0.506i)8-s − 1.98·9-s + (−0.151 + 0.420i)10-s − 0.301i·11-s + (1.10 − 1.32i)12-s − 1.08·13-s + (−1.60 − 0.581i)14-s + 0.773i·15-s + (0.183 + 0.982i)16-s + 1.22·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.638 + 0.769i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $-0.638 + 0.769i$
Analytic conductor: \(5.99456\)
Root analytic conductor: \(2.44838\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1),\ -0.638 + 0.769i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.442972 - 0.943621i\)
\(L(\frac12)\) \(\approx\) \(0.442972 - 0.943621i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.679 - 1.88i)T \)
5 \( 1 - 2.23T \)
11 \( 1 + 3.31iT \)
good3 \( 1 - 5.18iT - 9T^{2} \)
7 \( 1 - 11.9iT - 49T^{2} \)
13 \( 1 + 14.0T + 169T^{2} \)
17 \( 1 - 20.7T + 289T^{2} \)
19 \( 1 - 0.389iT - 361T^{2} \)
23 \( 1 + 7.30iT - 529T^{2} \)
29 \( 1 - 39.6T + 841T^{2} \)
31 \( 1 + 24.7iT - 961T^{2} \)
37 \( 1 - 28.5T + 1.36e3T^{2} \)
41 \( 1 + 41.8T + 1.68e3T^{2} \)
43 \( 1 + 2.07iT - 1.84e3T^{2} \)
47 \( 1 - 87.2iT - 2.20e3T^{2} \)
53 \( 1 + 103.T + 2.80e3T^{2} \)
59 \( 1 - 44.6iT - 3.48e3T^{2} \)
61 \( 1 - 61.7T + 3.72e3T^{2} \)
67 \( 1 - 36.1iT - 4.48e3T^{2} \)
71 \( 1 - 98.8iT - 5.04e3T^{2} \)
73 \( 1 + 10.3T + 5.32e3T^{2} \)
79 \( 1 + 59.9iT - 6.24e3T^{2} \)
83 \( 1 - 23.4iT - 6.88e3T^{2} \)
89 \( 1 + 55.8T + 7.92e3T^{2} \)
97 \( 1 - 86.7T + 9.40e3T^{2} \)
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

−12.64093458875625216282069529380, −11.51836812371565218990386272068, −10.18907421660070387117111027680, −9.662306464215124713559059575501, −8.937104592009706986209676693936, −8.044659853525609717542017005903, −6.15173150188596364990073198861, −5.40984931305619557418815040539, −4.62967854185190857137880263100, −2.87078879205141979280058548253, 0.66518569454909269345638266438, 1.75134699295395282826293209549, 3.21219150045635339160283738689, 4.94257319933837260730487973717, 6.72302998230954886110955283338, 7.48619615056692915309294567427, 8.202467614637206747693751705057, 9.756149210580271326778245454639, 10.45497571429484728704599616090, 11.67128569091152390441299044241

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