Properties

Label 2-1800-40.19-c0-0-1
Degree $2$
Conductor $1800$
Sign $0.447 - 0.894i$
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  + i·2-s − 4-s i·8-s + 11-s + 16-s i·17-s + 19-s + i·22-s + i·32-s + 34-s + i·38-s + 41-s + 2i·43-s − 44-s − 49-s + ⋯
L(s)  = 1  + i·2-s − 4-s i·8-s + 11-s + 16-s i·17-s + 19-s + i·22-s + i·32-s + 34-s + i·38-s + 41-s + 2i·43-s − 44-s − 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(0.898317\)
Root analytic conductor: \(0.947795\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (1099, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :0),\ 0.447 - 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.097653936\)
\(L(\frac12)\) \(\approx\) \(1.097653936\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.522427778975034796057817967191, −8.775576639026423808114104785420, −7.908410925959599197103503834839, −7.18795282510836903390806698690, −6.51105910546158158543828620353, −5.66682520410133423496118719312, −4.83169195739333518555481908427, −3.99185489963493349396478221504, −2.95654449017666676530158671064, −1.14818931593202013345070836726, 1.17736854813761512151093735038, 2.25594644678019811115267924317, 3.48866509072144653531176499563, 4.04969398771955654931045658331, 5.12558399548621274911847246335, 5.95043427263604448866412673948, 6.98562482909292943684520635666, 7.997213166894221633581959269714, 8.758361273880077137412131551180, 9.423014382568503879193719248955

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