↑

Properties

Label 2-1620-15.14-c2-0-30
Degree $2$
Conductor $1620$
Sign $0.998 + 0.0622i$
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  + (4.99 + 0.311i)5-s − 4.65i·7-s + 12.0i·11-s − 8.46i·13-s − 1.25·17-s + 16.4·19-s + 2.38·23-s + (24.8 + 3.10i)25-s + 23.9i·29-s − 5.25·31-s + (1.44 − 23.2i)35-s + 25.1i·37-s − 29.8i·41-s − 25.5i·43-s + 64.9·47-s + ⋯
L(s)  = 1  + (0.998 + 0.0622i)5-s − 0.665i·7-s + 1.09i·11-s − 0.651i·13-s − 0.0736·17-s + 0.867·19-s + 0.103·23-s + (0.992 + 0.124i)25-s + 0.827i·29-s − 0.169·31-s + (0.0413 − 0.663i)35-s + 0.680i·37-s − 0.727i·41-s − 0.593i·43-s + 1.38·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0622i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.998 + 0.0622i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ 0.998 + 0.0622i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.577241538\)
\(L(\frac12)\) \(\approx\) \(2.577241538\)
\(L(2)\) 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 \( 1 + (-4.99 - 0.311i)T \)
good7 \( 1 + 4.65iT - 49T^{2} \)
11 \( 1 - 12.0iT - 121T^{2} \)
13 \( 1 + 8.46iT - 169T^{2} \)
17 \( 1 + 1.25T + 289T^{2} \)
19 \( 1 - 16.4T + 361T^{2} \)
23 \( 1 - 2.38T + 529T^{2} \)
29 \( 1 - 23.9iT - 841T^{2} \)
31 \( 1 + 5.25T + 961T^{2} \)
37 \( 1 - 25.1iT - 1.36e3T^{2} \)
41 \( 1 + 29.8iT - 1.68e3T^{2} \)
43 \( 1 + 25.5iT - 1.84e3T^{2} \)
47 \( 1 - 64.9T + 2.20e3T^{2} \)
53 \( 1 - 71.4T + 2.80e3T^{2} \)
59 \( 1 - 18.5iT - 3.48e3T^{2} \)
61 \( 1 - 19.1T + 3.72e3T^{2} \)
67 \( 1 + 81.0iT - 4.48e3T^{2} \)
71 \( 1 + 21.9iT - 5.04e3T^{2} \)
73 \( 1 - 109. iT - 5.32e3T^{2} \)
79 \( 1 + 134.T + 6.24e3T^{2} \)
83 \( 1 - 15.6T + 6.88e3T^{2} \)
89 \( 1 - 79.4iT - 7.92e3T^{2} \)
97 \( 1 + 130. iT - 9.40e3T^{2} \)
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

−9.295161834015531903945206293521, −8.522094955508686469256526483208, −7.25608844167340650221140107030, −7.08032595723229936128149279412, −5.82565908317094303851224447979, −5.19169893114796396213988878626, −4.21927036846170217252447801741, −3.08283440104449388324890986735, −2.03049096937329001996356533845, −0.932926804504546800985565632612, 0.927844379830555406321255726040, 2.16301402637159745608810273094, 3.00102172345681382999000626323, 4.20217600965126975518876165653, 5.39443903318490965688421327524, 5.83368104766179536451675193078, 6.64540818492864568993723554664, 7.64205262499018919330275755405, 8.705153387093193517536685949313, 9.113626968073079794425387186499

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