Properties

Label 2-1900-1.1-c1-0-11
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  − 2.91·3-s − 2.91·7-s + 5.48·9-s + 0.598·11-s + 2.56·13-s − 2.91·17-s − 19-s + 8.48·21-s + 1.16·23-s − 7.22·27-s − 1.40·29-s + 7.88·31-s − 1.74·33-s + 6.53·37-s − 7.48·39-s + 9.48·41-s + 0.824·43-s − 6.05·47-s + 1.48·49-s + 8.48·51-s + 0.824·53-s + 2.91·57-s − 14.1·59-s − 1.88·61-s − 15.9·63-s − 5.11·67-s − 3.40·69-s + ⋯
L(s)  = 1  − 1.68·3-s − 1.10·7-s + 1.82·9-s + 0.180·11-s + 0.712·13-s − 0.706·17-s − 0.229·19-s + 1.85·21-s + 0.243·23-s − 1.39·27-s − 0.260·29-s + 1.41·31-s − 0.303·33-s + 1.07·37-s − 1.19·39-s + 1.48·41-s + 0.125·43-s − 0.883·47-s + 0.211·49-s + 1.18·51-s + 0.113·53-s + 0.385·57-s − 1.84·59-s − 0.240·61-s − 2.01·63-s − 0.624·67-s − 0.409·69-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 \( 1 + 2.91T + 3T^{2} \)
7 \( 1 + 2.91T + 7T^{2} \)
11 \( 1 - 0.598T + 11T^{2} \)
13 \( 1 - 2.56T + 13T^{2} \)
17 \( 1 + 2.91T + 17T^{2} \)
23 \( 1 - 1.16T + 23T^{2} \)
29 \( 1 + 1.40T + 29T^{2} \)
31 \( 1 - 7.88T + 31T^{2} \)
37 \( 1 - 6.53T + 37T^{2} \)
41 \( 1 - 9.48T + 41T^{2} \)
43 \( 1 - 0.824T + 43T^{2} \)
47 \( 1 + 6.05T + 47T^{2} \)
53 \( 1 - 0.824T + 53T^{2} \)
59 \( 1 + 14.1T + 59T^{2} \)
61 \( 1 + 1.88T + 61T^{2} \)
67 \( 1 + 5.11T + 67T^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 + 1.03T + 73T^{2} \)
79 \( 1 + 15.3T + 79T^{2} \)
83 \( 1 - 15.7T + 83T^{2} \)
89 \( 1 - 15.4T + 89T^{2} \)
97 \( 1 + 7.70T + 97T^{2} \)
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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

−9.065544860841358223931970001410, −7.88375002073260156012949291420, −6.87381590083381052954622797379, −6.24324375058328167995053844109, −5.93443749351693622668376734064, −4.76841061830974078263335124269, −4.08981849159772921019522733733, −2.84553871752581651325104418607, −1.20811503734359560501994971125, 0, 1.20811503734359560501994971125, 2.84553871752581651325104418607, 4.08981849159772921019522733733, 4.76841061830974078263335124269, 5.93443749351693622668376734064, 6.24324375058328167995053844109, 6.87381590083381052954622797379, 7.88375002073260156012949291420, 9.065544860841358223931970001410

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