↑

Properties

Label 2-1620-45.29-c2-0-39
Degree $2$
Conductor $1620$
Sign $0.999 + 0.0244i$
Analytic cond. $44.1418$
Root an. cond. $6.64392$
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  + (4.58 − 2.00i)5-s + (9.28 + 5.36i)7-s + (10.5 + 6.09i)11-s + (20.2 − 11.6i)13-s + 1.67·17-s + 0.234·19-s + (−9.93 − 17.2i)23-s + (16.9 − 18.3i)25-s + (29.5 + 17.0i)29-s + (9.53 + 16.5i)31-s + (53.2 + 5.97i)35-s − 32.7i·37-s + (−4.57 + 2.63i)41-s + (−45.8 − 26.4i)43-s + (−26.8 + 46.5i)47-s + ⋯
L(s)  = 1  + (0.916 − 0.400i)5-s + (1.32 + 0.766i)7-s + (0.959 + 0.553i)11-s + (1.55 − 0.899i)13-s + 0.0987·17-s + 0.0123·19-s + (−0.431 − 0.747i)23-s + (0.679 − 0.733i)25-s + (1.01 + 0.587i)29-s + (0.307 + 0.532i)31-s + (1.52 + 0.170i)35-s − 0.884i·37-s + (−0.111 + 0.0643i)41-s + (−1.06 − 0.616i)43-s + (−0.571 + 0.989i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0244i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0244i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.999 + 0.0244i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ 0.999 + 0.0244i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.437455862\)
\(L(\frac12)\) \(\approx\) \(3.437455862\)
\(L(2)\) 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 \( 1 + (-4.58 + 2.00i)T \)
good7 \( 1 + (-9.28 - 5.36i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-10.5 - 6.09i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-20.2 + 11.6i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 1.67T + 289T^{2} \)
19 \( 1 - 0.234T + 361T^{2} \)
23 \( 1 + (9.93 + 17.2i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-29.5 - 17.0i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-9.53 - 16.5i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 32.7iT - 1.36e3T^{2} \)
41 \( 1 + (4.57 - 2.63i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (45.8 + 26.4i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (26.8 - 46.5i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 84.6T + 2.80e3T^{2} \)
59 \( 1 + (76.4 - 44.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-32.7 + 56.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (86.1 - 49.7i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 36.8iT - 5.04e3T^{2} \)
73 \( 1 - 79.0iT - 5.32e3T^{2} \)
79 \( 1 + (17.6 - 30.6i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-13.9 + 24.1i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 152. iT - 7.92e3T^{2} \)
97 \( 1 + (-84.0 - 48.5i)T + (4.70e3 + 8.14e3i)T^{2} \)
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

−8.922657084359277563245866821039, −8.619653908594434654396592957077, −7.85182873695804205828086086386, −6.54138355776885093698344952372, −5.95774362688555622592627337250, −5.09474774640817420355117702763, −4.39461535354934330157625871665, −3.06673242682226801536880082920, −1.80966210262704832970127002087, −1.20341187801446996272838074614, 1.25094450445169527977594661963, 1.71881642924458199926522758209, 3.28581429374927791363833491230, 4.17282594699870395873638316459, 5.04434848865852942278765711748, 6.28916858988688032730667318194, 6.46490625575387505724555006126, 7.71817817538014497077617481389, 8.428571226918719299107759021707, 9.181022413250645277517920267944

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