Properties

Label 2-1900-1.1-c1-0-8
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.286·3-s + 0.286·7-s − 2.91·9-s + 4.26·11-s + 3.20·13-s + 0.286·17-s − 19-s + 0.0820·21-s + 0.936·23-s − 1.69·27-s + 2.26·29-s − 4.18·31-s + 1.22·33-s + 8.67·37-s + 0.917·39-s + 1.08·41-s + 4.42·43-s − 0.759·47-s − 6.91·49-s + 0.0820·51-s + 4.42·53-s − 0.286·57-s − 4.70·59-s + 10.1·61-s − 0.835·63-s + 9.82·67-s + 0.268·69-s + ⋯
L(s)  = 1  + 0.165·3-s + 0.108·7-s − 0.972·9-s + 1.28·11-s + 0.888·13-s + 0.0694·17-s − 0.229·19-s + 0.0179·21-s + 0.195·23-s − 0.326·27-s + 0.421·29-s − 0.751·31-s + 0.212·33-s + 1.42·37-s + 0.146·39-s + 0.168·41-s + 0.675·43-s − 0.110·47-s − 0.988·49-s + 0.0114·51-s + 0.608·53-s − 0.0379·57-s − 0.611·59-s + 1.30·61-s − 0.105·63-s + 1.20·67-s + 0.0322·69-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: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.942389140\)
\(L(\frac12)\) \(\approx\) \(1.942389140\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.286T + 3T^{2} \)
7 \( 1 - 0.286T + 7T^{2} \)
11 \( 1 - 4.26T + 11T^{2} \)
13 \( 1 - 3.20T + 13T^{2} \)
17 \( 1 - 0.286T + 17T^{2} \)
23 \( 1 - 0.936T + 23T^{2} \)
29 \( 1 - 2.26T + 29T^{2} \)
31 \( 1 + 4.18T + 31T^{2} \)
37 \( 1 - 8.67T + 37T^{2} \)
41 \( 1 - 1.08T + 41T^{2} \)
43 \( 1 - 4.42T + 43T^{2} \)
47 \( 1 + 0.759T + 47T^{2} \)
53 \( 1 - 4.42T + 53T^{2} \)
59 \( 1 + 4.70T + 59T^{2} \)
61 \( 1 - 10.1T + 61T^{2} \)
67 \( 1 - 9.82T + 67T^{2} \)
71 \( 1 - 4.83T + 71T^{2} \)
73 \( 1 - 10.4T + 73T^{2} \)
79 \( 1 - 5.10T + 79T^{2} \)
83 \( 1 - 15.7T + 83T^{2} \)
89 \( 1 + 9.75T + 89T^{2} \)
97 \( 1 + 9.61T + 97T^{2} \)
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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

−9.140879389235116038272545218772, −8.501584345334692094325451264357, −7.80931231480898487064626819310, −6.67391065364349984203915423216, −6.14042138886561357707251506789, −5.24356054433342705557098206047, −4.10668227728553220784954911106, −3.40982862762143731522495822904, −2.25951248532388438784273882041, −0.972027755179263679599206240816, 0.972027755179263679599206240816, 2.25951248532388438784273882041, 3.40982862762143731522495822904, 4.10668227728553220784954911106, 5.24356054433342705557098206047, 6.14042138886561357707251506789, 6.67391065364349984203915423216, 7.80931231480898487064626819310, 8.501584345334692094325451264357, 9.140879389235116038272545218772

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