Properties

Label 2-1800-1.1-c1-0-4
Degree $2$
Conductor $1800$
Sign $1$
Analytic cond. $14.3730$
Root an. cond. $3.79118$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s − 11-s − 4·13-s + 5·17-s + 19-s − 2·23-s + 8·29-s + 10·31-s + 6·37-s + 3·41-s − 4·43-s + 4·47-s − 3·49-s + 6·53-s − 8·59-s + 10·61-s + 67-s + 12·71-s − 3·73-s + 2·77-s + 6·79-s − 13·83-s + 9·89-s + 8·91-s + 14·97-s − 6·101-s − 4·103-s + ⋯
L(s)  = 1  − 0.755·7-s − 0.301·11-s − 1.10·13-s + 1.21·17-s + 0.229·19-s − 0.417·23-s + 1.48·29-s + 1.79·31-s + 0.986·37-s + 0.468·41-s − 0.609·43-s + 0.583·47-s − 3/7·49-s + 0.824·53-s − 1.04·59-s + 1.28·61-s + 0.122·67-s + 1.42·71-s − 0.351·73-s + 0.227·77-s + 0.675·79-s − 1.42·83-s + 0.953·89-s + 0.838·91-s + 1.42·97-s − 0.597·101-s − 0.394·103-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(14.3730\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.460240514\)
\(L(\frac12)\) \(\approx\) \(1.460240514\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 13 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−9.549345032844968179219680675990, −8.351506683319880487851775121042, −7.77075401088767705375118610734, −6.86378396127984906651881332641, −6.12495822792309039345594276838, −5.20481348804422316781252563482, −4.36172478777594960027231636860, −3.18472537975480801189708163896, −2.47956256564960138993576183683, −0.819426402868476389338501487291, 0.819426402868476389338501487291, 2.47956256564960138993576183683, 3.18472537975480801189708163896, 4.36172478777594960027231636860, 5.20481348804422316781252563482, 6.12495822792309039345594276838, 6.86378396127984906651881332641, 7.77075401088767705375118610734, 8.351506683319880487851775121042, 9.549345032844968179219680675990

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