Properties

Label 2-1900-1.1-c1-0-24
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.732·3-s − 2·7-s − 2.46·9-s + 3.46·11-s − 0.732·13-s − 3.46·17-s + 19-s − 1.46·21-s − 3.46·23-s − 4·27-s − 3.46·29-s + 5.46·31-s + 2.53·33-s − 3.26·37-s − 0.535·39-s − 6·41-s − 8.92·43-s + 0.928·47-s − 3·49-s − 2.53·51-s + 7.26·53-s + 0.732·57-s − 6.92·59-s − 8.39·61-s + 4.92·63-s − 3.26·67-s − 2.53·69-s + ⋯
L(s)  = 1  + 0.422·3-s − 0.755·7-s − 0.821·9-s + 1.04·11-s − 0.203·13-s − 0.840·17-s + 0.229·19-s − 0.319·21-s − 0.722·23-s − 0.769·27-s − 0.643·29-s + 0.981·31-s + 0.441·33-s − 0.537·37-s − 0.0858·39-s − 0.937·41-s − 1.36·43-s + 0.135·47-s − 0.428·49-s − 0.355·51-s + 0.998·53-s + 0.0969·57-s − 0.901·59-s − 1.07·61-s + 0.620·63-s − 0.399·67-s − 0.305·69-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 \( 1 - 0.732T + 3T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 - 3.46T + 11T^{2} \)
13 \( 1 + 0.732T + 13T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
23 \( 1 + 3.46T + 23T^{2} \)
29 \( 1 + 3.46T + 29T^{2} \)
31 \( 1 - 5.46T + 31T^{2} \)
37 \( 1 + 3.26T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 8.92T + 43T^{2} \)
47 \( 1 - 0.928T + 47T^{2} \)
53 \( 1 - 7.26T + 53T^{2} \)
59 \( 1 + 6.92T + 59T^{2} \)
61 \( 1 + 8.39T + 61T^{2} \)
67 \( 1 + 3.26T + 67T^{2} \)
71 \( 1 + 9.46T + 71T^{2} \)
73 \( 1 - 7.46T + 73T^{2} \)
79 \( 1 + 10.9T + 79T^{2} \)
83 \( 1 - 3.46T + 83T^{2} \)
89 \( 1 + 8.53T + 89T^{2} \)
97 \( 1 + 14.5T + 97T^{2} \)
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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

−8.818686164756380864525594806133, −8.243704919932355411372816219043, −7.16762095013087082325732917164, −6.45545676362758861993359922643, −5.76729417529343455527964321079, −4.61810810306989325090843943910, −3.65016321528989332605373908981, −2.91027176429740671157007530436, −1.76397749494652932195956149680, 0, 1.76397749494652932195956149680, 2.91027176429740671157007530436, 3.65016321528989332605373908981, 4.61810810306989325090843943910, 5.76729417529343455527964321079, 6.45545676362758861993359922643, 7.16762095013087082325732917164, 8.243704919932355411372816219043, 8.818686164756380864525594806133

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