Properties

Label 2-1900-1.1-c1-0-25
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  + 1.19·3-s + 1.19·7-s − 1.56·9-s − 5.86·11-s − 0.364·13-s + 1.19·17-s − 19-s + 1.43·21-s − 8.23·23-s − 5.46·27-s − 7.86·29-s + 7.30·31-s − 7.03·33-s + 7.13·37-s − 0.436·39-s + 2.43·41-s − 7.39·43-s − 13.7·47-s − 5.56·49-s + 1.43·51-s − 7.39·53-s − 1.19·57-s + 12.8·59-s − 1.30·61-s − 1.87·63-s + 11.9·67-s − 9.86·69-s + ⋯
L(s)  = 1  + 0.692·3-s + 0.453·7-s − 0.521·9-s − 1.76·11-s − 0.101·13-s + 0.290·17-s − 0.229·19-s + 0.313·21-s − 1.71·23-s − 1.05·27-s − 1.46·29-s + 1.31·31-s − 1.22·33-s + 1.17·37-s − 0.0699·39-s + 0.380·41-s − 1.12·43-s − 1.99·47-s − 0.794·49-s + 0.201·51-s − 1.01·53-s − 0.158·57-s + 1.67·59-s − 0.166·61-s − 0.236·63-s + 1.45·67-s − 1.18·69-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 \( 1 - 1.19T + 3T^{2} \)
7 \( 1 - 1.19T + 7T^{2} \)
11 \( 1 + 5.86T + 11T^{2} \)
13 \( 1 + 0.364T + 13T^{2} \)
17 \( 1 - 1.19T + 17T^{2} \)
23 \( 1 + 8.23T + 23T^{2} \)
29 \( 1 + 7.86T + 29T^{2} \)
31 \( 1 - 7.30T + 31T^{2} \)
37 \( 1 - 7.13T + 37T^{2} \)
41 \( 1 - 2.43T + 41T^{2} \)
43 \( 1 + 7.39T + 43T^{2} \)
47 \( 1 + 13.7T + 47T^{2} \)
53 \( 1 + 7.39T + 53T^{2} \)
59 \( 1 - 12.8T + 59T^{2} \)
61 \( 1 + 1.30T + 61T^{2} \)
67 \( 1 - 11.9T + 67T^{2} \)
71 \( 1 - 2.12T + 71T^{2} \)
73 \( 1 - 2.50T + 73T^{2} \)
79 \( 1 + 7.74T + 79T^{2} \)
83 \( 1 + 3.02T + 83T^{2} \)
89 \( 1 + 5.68T + 89T^{2} \)
97 \( 1 - 1.09T + 97T^{2} \)
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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

−8.551461093160894423181193566992, −7.994359911729091262771557118885, −7.71812629623321670961146813273, −6.37252904337519794972227129130, −5.53337862469520074919136207588, −4.78877237873351340495902590237, −3.65467397697074916693274332468, −2.71064648842484936641733728989, −1.95437803332624454046645372862, 0, 1.95437803332624454046645372862, 2.71064648842484936641733728989, 3.65467397697074916693274332468, 4.78877237873351340495902590237, 5.53337862469520074919136207588, 6.37252904337519794972227129130, 7.71812629623321670961146813273, 7.994359911729091262771557118885, 8.551461093160894423181193566992

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