↑

Properties

Label 2-1620-9.4-c3-0-18
Degree $2$
Conductor $1620$
Sign $0.766 - 0.642i$
Analytic cond. $95.5830$
Root an. cond. $9.77666$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.5 − 4.33i)5-s + (−0.474 − 0.822i)7-s + (31.3 + 54.2i)11-s + (−3.47 + 6.01i)13-s + 103.·17-s + 73.8·19-s + (43.0 − 74.6i)23-s + (−12.5 − 21.6i)25-s + (−27.7 − 48.1i)29-s + (−99.9 + 173. i)31-s − 4.74·35-s − 18.9·37-s + (−28.6 + 49.6i)41-s + (−148. − 256. i)43-s + (28.7 + 49.8i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (−0.0256 − 0.0444i)7-s + (0.858 + 1.48i)11-s + (−0.0741 + 0.128i)13-s + 1.47·17-s + 0.891·19-s + (0.390 − 0.676i)23-s + (−0.100 − 0.173i)25-s + (−0.177 − 0.308i)29-s + (−0.578 + 1.00i)31-s − 0.0229·35-s − 0.0840·37-s + (−0.109 + 0.189i)41-s + (−0.524 − 0.909i)43-s + (0.0892 + 0.154i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(95.5830\)
Root analytic conductor: \(9.77666\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :3/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.515815866\)
\(L(\frac12)\) \(\approx\) \(2.515815866\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.5 + 4.33i)T \)
good7 \( 1 + (0.474 + 0.822i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-31.3 - 54.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (3.47 - 6.01i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 103.T + 4.91e3T^{2} \)
19 \( 1 - 73.8T + 6.85e3T^{2} \)
23 \( 1 + (-43.0 + 74.6i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (27.7 + 48.1i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (99.9 - 173. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 18.9T + 5.06e4T^{2} \)
41 \( 1 + (28.6 - 49.6i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (148. + 256. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-28.7 - 49.8i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 672.T + 1.48e5T^{2} \)
59 \( 1 + (273. - 473. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-395. - 685. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (315. - 546. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 102.T + 3.57e5T^{2} \)
73 \( 1 + 200.T + 3.89e5T^{2} \)
79 \( 1 + (-332. - 576. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-673. - 1.16e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 12.6T + 7.04e5T^{2} \)
97 \( 1 + (-318. - 551. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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

−9.237845995318971774595678920207, −8.388794045274929826878090834606, −7.35922911638290548524871745995, −6.91062907169154497979640630347, −5.76073037807341025377299284570, −5.00341712556074199317382000139, −4.15418770964063357486388464421, −3.14187768928383184646489084073, −1.85637039294700274950010712844, −1.01348053581673737073261856465, 0.65213445867591631333707162639, 1.65661640727364355812307616483, 3.30327528166789611901026515796, 3.38782212027835013437355620159, 4.94068743274375283232735401863, 5.82555875107068850588899274439, 6.33391501512282577412235764439, 7.49008328587669574369536505953, 8.007495107564295304008700788415, 9.148023011035933240403365798550

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