Properties

Label 2-1800-200.173-c0-0-1
Degree $2$
Conductor $1800$
Sign $-0.827 + 0.562i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.987 − 0.156i)2-s + (0.951 + 0.309i)4-s + (−0.156 + 0.987i)5-s + (−1.26 + 1.26i)7-s + (−0.891 − 0.453i)8-s + (0.309 − 0.951i)10-s + (−1.16 − 1.59i)11-s + (1.44 − 1.04i)14-s + (0.809 + 0.587i)16-s + (−0.453 + 0.891i)20-s + (0.896 + 1.76i)22-s + (−0.951 − 0.309i)25-s + (−1.58 + 0.809i)28-s + (−0.437 + 1.34i)29-s + (−0.587 − 1.80i)31-s + (−0.707 − 0.707i)32-s + ⋯
L(s)  = 1  + (−0.987 − 0.156i)2-s + (0.951 + 0.309i)4-s + (−0.156 + 0.987i)5-s + (−1.26 + 1.26i)7-s + (−0.891 − 0.453i)8-s + (0.309 − 0.951i)10-s + (−1.16 − 1.59i)11-s + (1.44 − 1.04i)14-s + (0.809 + 0.587i)16-s + (−0.453 + 0.891i)20-s + (0.896 + 1.76i)22-s + (−0.951 − 0.309i)25-s + (−1.58 + 0.809i)28-s + (−0.437 + 1.34i)29-s + (−0.587 − 1.80i)31-s + (−0.707 − 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.827 + 0.562i$
Analytic conductor: \(0.898317\)
Root analytic conductor: \(0.947795\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (973, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :0),\ -0.827 + 0.562i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.008754437862\)
\(L(\frac12)\) \(\approx\) \(0.008754437862\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.987 + 0.156i)T \)
3 \( 1 \)
5 \( 1 + (0.156 - 0.987i)T \)
good7 \( 1 + (1.26 - 1.26i)T - iT^{2} \)
11 \( 1 + (1.16 + 1.59i)T + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.951 - 0.309i)T^{2} \)
17 \( 1 + (0.587 + 0.809i)T^{2} \)
19 \( 1 + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (0.951 + 0.309i)T^{2} \)
29 \( 1 + (0.437 - 1.34i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.587 + 1.80i)T + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.951 + 0.309i)T^{2} \)
41 \( 1 + (0.309 + 0.951i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-0.587 + 0.809i)T^{2} \)
53 \( 1 + (0.533 + 1.04i)T + (-0.587 + 0.809i)T^{2} \)
59 \( 1 + (-0.253 - 0.183i)T + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T^{2} \)
67 \( 1 + (-0.587 - 0.809i)T^{2} \)
71 \( 1 + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (1.39 + 0.221i)T + (0.951 + 0.309i)T^{2} \)
79 \( 1 + (1.11 + 0.363i)T + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.550 + 0.280i)T + (0.587 + 0.809i)T^{2} \)
89 \( 1 + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (0.412 + 0.809i)T + (-0.587 + 0.809i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.132884664430485985067361830159, −8.485937410925689267199773657012, −7.69508468744806378678002037072, −6.84305946488816248947525670871, −5.96829857430217243638907080309, −5.64966853864704235409588798072, −3.50547463994565677586348641066, −3.00492956398023495268618850838, −2.26596409322612176906348974025, −0.008554795011706307577689978402, 1.47994800551165332851957846553, 2.76754724568169578823778752196, 3.99274673574568988243870348259, 4.93509476241500222471179710987, 5.94042198378166192970032881371, 7.02001696738273066511345310831, 7.39418140394358267824701489443, 8.158745820909820608320733554552, 9.142043159353979060852439516576, 9.822543587830719140679850685102

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