Properties

Label 2-1900-1.1-c1-0-3
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  − 0.750·3-s − 0.872·7-s − 2.43·9-s − 1.11·11-s − 4.58·13-s + 2.96·17-s + 19-s + 0.654·21-s − 3.83·23-s + 4.07·27-s + 7.56·29-s + 9.56·31-s + 0.832·33-s − 0.750·37-s + 3.43·39-s + 8.87·41-s + 11.5·43-s − 8.53·47-s − 6.23·49-s − 2.22·51-s − 8.17·53-s − 0.750·57-s + 4.65·59-s + 0.889·61-s + 2.12·63-s − 2.59·67-s + 2.87·69-s + ⋯
L(s)  = 1  − 0.433·3-s − 0.329·7-s − 0.812·9-s − 0.334·11-s − 1.27·13-s + 0.717·17-s + 0.229·19-s + 0.142·21-s − 0.799·23-s + 0.784·27-s + 1.40·29-s + 1.71·31-s + 0.144·33-s − 0.123·37-s + 0.550·39-s + 1.38·41-s + 1.75·43-s − 1.24·47-s − 0.891·49-s − 0.310·51-s − 1.12·53-s − 0.0993·57-s + 0.605·59-s + 0.113·61-s + 0.268·63-s − 0.316·67-s + 0.346·69-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

\(L(1)\) \(\approx\) \(1.069859181\)
\(L(\frac12)\) \(\approx\) \(1.069859181\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.750T + 3T^{2} \)
7 \( 1 + 0.872T + 7T^{2} \)
11 \( 1 + 1.11T + 11T^{2} \)
13 \( 1 + 4.58T + 13T^{2} \)
17 \( 1 - 2.96T + 17T^{2} \)
23 \( 1 + 3.83T + 23T^{2} \)
29 \( 1 - 7.56T + 29T^{2} \)
31 \( 1 - 9.56T + 31T^{2} \)
37 \( 1 + 0.750T + 37T^{2} \)
41 \( 1 - 8.87T + 41T^{2} \)
43 \( 1 - 11.5T + 43T^{2} \)
47 \( 1 + 8.53T + 47T^{2} \)
53 \( 1 + 8.17T + 53T^{2} \)
59 \( 1 - 4.65T + 59T^{2} \)
61 \( 1 - 0.889T + 61T^{2} \)
67 \( 1 + 2.59T + 67T^{2} \)
71 \( 1 - 7.34T + 71T^{2} \)
73 \( 1 - 11.8T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 3.83T + 83T^{2} \)
89 \( 1 - 1.34T + 89T^{2} \)
97 \( 1 + 8.17T + 97T^{2} \)
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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

−9.423407520489386876278378691600, −8.223210037870443634319751780392, −7.81278457547149526470197316889, −6.69296852693287183690595733801, −6.04713906802971174259037331141, −5.18428777430704076802784457896, −4.47230674440475733427138606442, −3.13215551145543620606969761439, −2.43139520037091939997664557467, −0.68904144651399685866559148777, 0.68904144651399685866559148777, 2.43139520037091939997664557467, 3.13215551145543620606969761439, 4.47230674440475733427138606442, 5.18428777430704076802784457896, 6.04713906802971174259037331141, 6.69296852693287183690595733801, 7.81278457547149526470197316889, 8.223210037870443634319751780392, 9.423407520489386876278378691600

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