Properties

Label 2-1900-1.1-c1-0-1
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  − 1.62·3-s − 4.74·7-s − 0.367·9-s + 4.48·11-s − 0.843·13-s − 5.52·17-s + 19-s + 7.69·21-s + 0.779·23-s + 5.46·27-s − 10.6·29-s − 8.65·31-s − 7.26·33-s − 1.62·37-s + 1.36·39-s + 4.73·41-s + 9.67·43-s − 3.18·47-s + 15.5·49-s + 8.96·51-s + 6.17·53-s − 1.62·57-s + 11.6·59-s + 6.48·61-s + 1.74·63-s + 14.8·67-s − 1.26·69-s + ⋯
L(s)  = 1  − 0.936·3-s − 1.79·7-s − 0.122·9-s + 1.35·11-s − 0.233·13-s − 1.33·17-s + 0.229·19-s + 1.67·21-s + 0.162·23-s + 1.05·27-s − 1.97·29-s − 1.55·31-s − 1.26·33-s − 0.266·37-s + 0.219·39-s + 0.739·41-s + 1.47·43-s − 0.464·47-s + 2.21·49-s + 1.25·51-s + 0.848·53-s − 0.214·57-s + 1.52·59-s + 0.829·61-s + 0.219·63-s + 1.81·67-s − 0.152·69-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

\(L(1)\) \(\approx\) \(0.6378818216\)
\(L(\frac12)\) \(\approx\) \(0.6378818216\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.62T + 3T^{2} \)
7 \( 1 + 4.74T + 7T^{2} \)
11 \( 1 - 4.48T + 11T^{2} \)
13 \( 1 + 0.843T + 13T^{2} \)
17 \( 1 + 5.52T + 17T^{2} \)
23 \( 1 - 0.779T + 23T^{2} \)
29 \( 1 + 10.6T + 29T^{2} \)
31 \( 1 + 8.65T + 31T^{2} \)
37 \( 1 + 1.62T + 37T^{2} \)
41 \( 1 - 4.73T + 41T^{2} \)
43 \( 1 - 9.67T + 43T^{2} \)
47 \( 1 + 3.18T + 47T^{2} \)
53 \( 1 - 6.17T + 53T^{2} \)
59 \( 1 - 11.6T + 59T^{2} \)
61 \( 1 - 6.48T + 61T^{2} \)
67 \( 1 - 14.8T + 67T^{2} \)
71 \( 1 - 0.303T + 71T^{2} \)
73 \( 1 + 10.0T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 0.779T + 83T^{2} \)
89 \( 1 + 5.69T + 89T^{2} \)
97 \( 1 - 6.17T + 97T^{2} \)
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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

−9.258511744535299582162002244154, −8.779309504631923296370568401122, −7.18169834560311729195130255998, −6.82319593493255798845869425117, −6.00569609876773148642158085481, −5.48930011817670878195103570678, −4.14016570498397586212028156149, −3.50519090073751939483112350586, −2.24622620408940308031995104146, −0.53756157683053079461003563806, 0.53756157683053079461003563806, 2.24622620408940308031995104146, 3.50519090073751939483112350586, 4.14016570498397586212028156149, 5.48930011817670878195103570678, 6.00569609876773148642158085481, 6.82319593493255798845869425117, 7.18169834560311729195130255998, 8.779309504631923296370568401122, 9.258511744535299582162002244154

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