Properties

Label 2-819-1.1-c1-0-15
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 + 5-s + 7-s + 2·10-s + 2·11-s + 13-s + 2·14-s − 4·16-s + 4·17-s + 3·19-s + 2·20-s + 4·22-s + 9·23-s − 4·25-s + 2·26-s + 2·28-s + 29-s − 5·31-s − 8·32-s + 8·34-s + 35-s − 8·37-s + 6·38-s − 6·41-s − 9·43-s + 4·44-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 0.447·5-s + 0.377·7-s + 0.632·10-s + 0.603·11-s + 0.277·13-s + 0.534·14-s − 16-s + 0.970·17-s + 0.688·19-s + 0.447·20-s + 0.852·22-s + 1.87·23-s − 4/5·25-s + 0.392·26-s + 0.377·28-s + 0.185·29-s − 0.898·31-s − 1.41·32-s + 1.37·34-s + 0.169·35-s − 1.31·37-s + 0.973·38-s − 0.937·41-s − 1.37·43-s + 0.603·44-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

\(L(1)\) \(\approx\) \(3.615147882\)
\(L(\frac12)\) \(\approx\) \(3.615147882\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 5 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−10.39106407242581279688311065672, −9.392431515365089468400672952772, −8.609580545226665465862939213661, −7.32071940346067656438365703279, −6.54249635829557594471463650102, −5.48555898600936918224764827719, −5.05576990864366814121710825740, −3.81057758241492887556514849122, −3.05793239424379990664179415786, −1.57788335674683088622292365244, 1.57788335674683088622292365244, 3.05793239424379990664179415786, 3.81057758241492887556514849122, 5.05576990864366814121710825740, 5.48555898600936918224764827719, 6.54249635829557594471463650102, 7.32071940346067656438365703279, 8.609580545226665465862939213661, 9.392431515365089468400672952772, 10.39106407242581279688311065672

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