Properties

Label 2-1800-1.1-c1-0-7
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

\(L(1)\) \(\approx\) \(1.959118232\)
\(L(\frac12)\) \(\approx\) \(1.959118232\)
\(L(\frac{3}{2})\) 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

−9.132092680695771174953740233683, −8.406424372567760315818180510871, −7.931548676274788519838470628634, −6.74970851104282883574216601300, −6.22064765023570818595861610948, −5.03102211092246474566104748762, −4.47573040070648122823436087963, −3.32538074522138155845712598448, −2.23448918468628852686878827752, −1.00769235920250604211457060594, 1.00769235920250604211457060594, 2.23448918468628852686878827752, 3.32538074522138155845712598448, 4.47573040070648122823436087963, 5.03102211092246474566104748762, 6.22064765023570818595861610948, 6.74970851104282883574216601300, 7.931548676274788519838470628634, 8.406424372567760315818180510871, 9.132092680695771174953740233683

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