Properties

Label 2-1900-1.1-c1-0-27
Degree $2$
Conductor $1900$
Sign $-1$
Analytic cond. $15.1715$
Root an. cond. $3.89507$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.18·3-s − 3.70·7-s + 1.76·9-s − 3.18·11-s + 1.94·13-s − 4.66·17-s + 19-s − 8.08·21-s − 5.23·23-s − 2.70·27-s − 1.66·29-s − 3.46·31-s − 6.94·33-s + 7.18·37-s + 4.23·39-s − 0.717·41-s + 1.36·43-s − 5.37·47-s + 6.71·49-s − 10.1·51-s − 0.0435·53-s + 2.18·57-s − 7.88·59-s − 4.98·61-s − 6.52·63-s − 14.5·67-s − 11.4·69-s + ⋯
L(s)  = 1  + 1.25·3-s − 1.39·7-s + 0.586·9-s − 0.959·11-s + 0.538·13-s − 1.13·17-s + 0.229·19-s − 1.76·21-s − 1.09·23-s − 0.520·27-s − 0.308·29-s − 0.622·31-s − 1.20·33-s + 1.18·37-s + 0.678·39-s − 0.112·41-s + 0.207·43-s − 0.784·47-s + 0.959·49-s − 1.42·51-s − 0.00597·53-s + 0.289·57-s − 1.02·59-s − 0.638·61-s − 0.821·63-s − 1.77·67-s − 1.37·69-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(15.1715\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 - T \)
good3 \( 1 - 2.18T + 3T^{2} \)
7 \( 1 + 3.70T + 7T^{2} \)
11 \( 1 + 3.18T + 11T^{2} \)
13 \( 1 - 1.94T + 13T^{2} \)
17 \( 1 + 4.66T + 17T^{2} \)
23 \( 1 + 5.23T + 23T^{2} \)
29 \( 1 + 1.66T + 29T^{2} \)
31 \( 1 + 3.46T + 31T^{2} \)
37 \( 1 - 7.18T + 37T^{2} \)
41 \( 1 + 0.717T + 41T^{2} \)
43 \( 1 - 1.36T + 43T^{2} \)
47 \( 1 + 5.37T + 47T^{2} \)
53 \( 1 + 0.0435T + 53T^{2} \)
59 \( 1 + 7.88T + 59T^{2} \)
61 \( 1 + 4.98T + 61T^{2} \)
67 \( 1 + 14.5T + 67T^{2} \)
71 \( 1 - 4.88T + 71T^{2} \)
73 \( 1 + 10.6T + 73T^{2} \)
79 \( 1 - 14.9T + 79T^{2} \)
83 \( 1 + 9.76T + 83T^{2} \)
89 \( 1 - 10.0T + 89T^{2} \)
97 \( 1 - 14.6T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.968857537820316991092286185252, −8.048985526449180672080670061654, −7.48039322145522093271554322005, −6.43873843379510083771439849346, −5.80232781463336422858395235642, −4.46532343682410600224227115559, −3.53063917703608566567030418224, −2.88780524972653745712949156724, −2.01822135036020651751822802961, 0, 2.01822135036020651751822802961, 2.88780524972653745712949156724, 3.53063917703608566567030418224, 4.46532343682410600224227115559, 5.80232781463336422858395235642, 6.43873843379510083771439849346, 7.48039322145522093271554322005, 8.048985526449180672080670061654, 8.968857537820316991092286185252

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