Properties

Label 2-819-1.1-c1-0-18
Degree $2$
Conductor $819$
Sign $1$
Analytic cond. $6.53974$
Root an. cond. $2.55729$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s + 3·5-s − 7-s + 6·10-s + 6·11-s − 13-s − 2·14-s − 4·16-s − 4·17-s + 5·19-s + 6·20-s + 12·22-s − 3·23-s + 4·25-s − 2·26-s − 2·28-s + 5·29-s − 3·31-s − 8·32-s − 8·34-s − 3·35-s − 4·37-s + 10·38-s + 6·41-s − 43-s + 12·44-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 1.34·5-s − 0.377·7-s + 1.89·10-s + 1.80·11-s − 0.277·13-s − 0.534·14-s − 16-s − 0.970·17-s + 1.14·19-s + 1.34·20-s + 2.55·22-s − 0.625·23-s + 4/5·25-s − 0.392·26-s − 0.377·28-s + 0.928·29-s − 0.538·31-s − 1.41·32-s − 1.37·34-s − 0.507·35-s − 0.657·37-s + 1.62·38-s + 0.937·41-s − 0.152·43-s + 1.80·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $1$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.855007959\)
\(L(\frac12)\) \(\approx\) \(3.855007959\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + T \)
13 \( 1 + T \)
good2 \( 1 - p T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−10.20319256030432890349268306800, −9.362259319494049279221007732110, −8.895694395064855641013305453407, −7.14095628036298532453823579927, −6.35409767437233023841602048217, −5.88871707105200974021405692252, −4.85246247359459828104814508190, −3.93866594968967426469354468549, −2.87743708330085578837178550046, −1.68695310650066648413239890440, 1.68695310650066648413239890440, 2.87743708330085578837178550046, 3.93866594968967426469354468549, 4.85246247359459828104814508190, 5.88871707105200974021405692252, 6.35409767437233023841602048217, 7.14095628036298532453823579927, 8.895694395064855641013305453407, 9.362259319494049279221007732110, 10.20319256030432890349268306800

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