Properties

Label 2-1800-200.133-c0-0-0
Degree $2$
Conductor $1800$
Sign $0.860 - 0.509i$
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  + (−0.453 + 0.891i)2-s + (−0.587 − 0.809i)4-s + (0.891 + 0.453i)5-s + (1.39 − 1.39i)7-s + (0.987 − 0.156i)8-s + (−0.809 + 0.587i)10-s + (−0.863 + 0.280i)11-s + (0.610 + 1.87i)14-s + (−0.309 + 0.951i)16-s + (−0.156 − 0.987i)20-s + (0.142 − 0.896i)22-s + (0.587 + 0.809i)25-s + (−1.95 − 0.309i)28-s + (1.14 − 0.831i)29-s + (−0.951 − 0.690i)31-s + (−0.707 − 0.707i)32-s + ⋯
L(s)  = 1  + (−0.453 + 0.891i)2-s + (−0.587 − 0.809i)4-s + (0.891 + 0.453i)5-s + (1.39 − 1.39i)7-s + (0.987 − 0.156i)8-s + (−0.809 + 0.587i)10-s + (−0.863 + 0.280i)11-s + (0.610 + 1.87i)14-s + (−0.309 + 0.951i)16-s + (−0.156 − 0.987i)20-s + (0.142 − 0.896i)22-s + (0.587 + 0.809i)25-s + (−1.95 − 0.309i)28-s + (1.14 − 0.831i)29-s + (−0.951 − 0.690i)31-s + (−0.707 − 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.860 - 0.509i$
Analytic conductor: \(0.898317\)
Root analytic conductor: \(0.947795\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (1333, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :0),\ 0.860 - 0.509i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.128739174\)
\(L(\frac12)\) \(\approx\) \(1.128739174\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.453 - 0.891i)T \)
3 \( 1 \)
5 \( 1 + (-0.891 - 0.453i)T \)
good7 \( 1 + (-1.39 + 1.39i)T - iT^{2} \)
11 \( 1 + (0.863 - 0.280i)T + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.587 + 0.809i)T^{2} \)
17 \( 1 + (0.951 - 0.309i)T^{2} \)
19 \( 1 + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (-0.587 - 0.809i)T^{2} \)
29 \( 1 + (-1.14 + 0.831i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.951 + 0.690i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.587 - 0.809i)T^{2} \)
41 \( 1 + (-0.809 - 0.587i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-0.951 - 0.309i)T^{2} \)
53 \( 1 + (0.297 - 1.87i)T + (-0.951 - 0.309i)T^{2} \)
59 \( 1 + (-0.550 + 1.69i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.809 - 0.587i)T^{2} \)
67 \( 1 + (-0.951 + 0.309i)T^{2} \)
71 \( 1 + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.642 - 1.26i)T + (-0.587 - 0.809i)T^{2} \)
79 \( 1 + (-1.11 - 1.53i)T + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (1.59 - 0.253i)T + (0.951 - 0.309i)T^{2} \)
89 \( 1 + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (0.0489 - 0.309i)T + (-0.951 - 0.309i)T^{2} \)
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

−9.633829988602541969685357977746, −8.520269969780796695104616670317, −7.81965166358317350108064205777, −7.29438020073792802797802480208, −6.51409662070287077383088486130, −5.51642415111702814042072818980, −4.85406099015248081162855923209, −4.01624611759756764051330815572, −2.29933680339848666106637461299, −1.19697753960302510219422727801, 1.46989387030480166228674942212, 2.22683579911069419632177290579, 3.06994509981760436281165850598, 4.67519957074371617925271394331, 5.14189762986454172655481889055, 5.89894711437990744011302639449, 7.30700927282015821633845003879, 8.376907009634480292237938633674, 8.581344404508217454273154859475, 9.307019706175796816464736363374

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