↑

Properties

Label 2-1800-360.203-c0-0-0
Degree $2$
Conductor $1800$
Sign $-0.737 - 0.675i$
Analytic cond. $0.898317$
Root an. cond. $0.947795$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.258 − 0.965i)3-s + (−0.866 − 0.499i)4-s + 6-s + (0.707 − 0.707i)8-s + (−0.866 + 0.499i)9-s + (−1.5 + 0.866i)11-s + (−0.258 + 0.965i)12-s + (0.500 + 0.866i)16-s + (−0.707 − 0.707i)17-s + (−0.258 − 0.965i)18-s + 2i·19-s + (−0.448 − 1.67i)22-s + (−0.866 − 0.5i)24-s + (0.707 + 0.707i)27-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.258 − 0.965i)3-s + (−0.866 − 0.499i)4-s + 6-s + (0.707 − 0.707i)8-s + (−0.866 + 0.499i)9-s + (−1.5 + 0.866i)11-s + (−0.258 + 0.965i)12-s + (0.500 + 0.866i)16-s + (−0.707 − 0.707i)17-s + (−0.258 − 0.965i)18-s + 2i·19-s + (−0.448 − 1.67i)22-s + (−0.866 − 0.5i)24-s + (0.707 + 0.707i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.737 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.737 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.737 - 0.675i$
Analytic conductor: \(0.898317\)
Root analytic conductor: \(0.947795\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (1643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :0),\ -0.737 - 0.675i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3851026171\)
\(L(\frac12)\) \(\approx\) \(0.3851026171\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 \)
good7 \( 1 + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.866 + 0.5i)T^{2} \)
17 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
19 \( 1 - 2iT - T^{2} \)
23 \( 1 + (0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \)
47 \( 1 + (0.866 + 0.5i)T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
89 \( 1 + 1.73T + T^{2} \)
97 \( 1 + (1.67 + 0.448i)T + (0.866 + 0.5i)T^{2} \)
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

−9.700301210395953580982480763425, −8.664905728787866886683826752837, −7.86228221240925168338566098997, −7.55213078664861039577749431005, −6.68935310591959125684012519567, −5.86031960313889540766277272320, −5.22129793153992868793028111658, −4.31118157152938583612947943735, −2.73882213614474802711523849326, −1.53070992825207934941872292849, 0.32159476653492231104791788035, 2.36556441924893165143430638966, 3.10752391706199167313223074670, 4.07072242480314610154038463632, 4.97679315853316708145964308805, 5.51392333070298981404760142905, 6.77726127651254815312285600465, 7.998890974741960906275544580476, 8.674517350938849413462727231800, 9.215301789877037783451146380519

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