Properties

Label 2-1900-19.18-c2-0-18
Degree $2$
Conductor $1900$
Sign $1$
Analytic cond. $51.7712$
Root an. cond. $7.19522$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 13.8·7-s + 9·9-s − 20.3·11-s − 18.9·17-s − 19·19-s + 30·23-s − 53.8·43-s + 86.5·47-s + 142.·49-s + 5.12·61-s − 124.·63-s + 112.·73-s + 281.·77-s + 81·81-s − 90·83-s − 183.·99-s − 102·101-s + 261.·119-s + ⋯
L(s)  = 1  − 1.97·7-s + 9-s − 1.85·11-s − 1.11·17-s − 19-s + 1.30·23-s − 1.25·43-s + 1.84·47-s + 2.90·49-s + 0.0839·61-s − 1.97·63-s + 1.53·73-s + 3.65·77-s + 81-s − 1.08·83-s − 1.85·99-s − 1.00·101-s + 2.19·119-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(51.7712\)
Root analytic conductor: \(7.19522\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (1101, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8590953258\)
\(L(\frac12)\) \(\approx\) \(0.8590953258\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 + 19T \)
good3 \( 1 - 9T^{2} \)
7 \( 1 + 13.8T + 49T^{2} \)
11 \( 1 + 20.3T + 121T^{2} \)
13 \( 1 - 169T^{2} \)
17 \( 1 + 18.9T + 289T^{2} \)
23 \( 1 - 30T + 529T^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 - 961T^{2} \)
37 \( 1 - 1.36e3T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 + 53.8T + 1.84e3T^{2} \)
47 \( 1 - 86.5T + 2.20e3T^{2} \)
53 \( 1 - 2.80e3T^{2} \)
59 \( 1 - 3.48e3T^{2} \)
61 \( 1 - 5.12T + 3.72e3T^{2} \)
67 \( 1 - 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 112.T + 5.32e3T^{2} \)
79 \( 1 - 6.24e3T^{2} \)
83 \( 1 + 90T + 6.88e3T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 - 9.40e3T^{2} \)
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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

−9.144666601389194797669518280595, −8.313038279620157081545421914458, −7.18101921575324547552914729216, −6.82191021843979775329222725351, −5.93421300898144336301304374404, −4.95948762092247983190963069769, −4.02024930766860252786133704575, −3.01080058983811147757540988983, −2.27491851266803815172964022623, −0.46845207038472105041304254503, 0.46845207038472105041304254503, 2.27491851266803815172964022623, 3.01080058983811147757540988983, 4.02024930766860252786133704575, 4.95948762092247983190963069769, 5.93421300898144336301304374404, 6.82191021843979775329222725351, 7.18101921575324547552914729216, 8.313038279620157081545421914458, 9.144666601389194797669518280595

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