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Properties

Label 2-819-91.90-c0-0-1
Degree $2$
Conductor $819$
Sign $1$
Analytic cond. $0.408734$
Root an. cond. $0.639323$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s + 7-s − 13-s + 16-s − 2·19-s − 25-s + 28-s + 2·31-s − 2·43-s + 49-s − 52-s + 64-s + 2·73-s − 2·76-s − 2·79-s − 91-s − 2·97-s − 100-s + 112-s + ⋯
L(s)  = 1  + 4-s + 7-s − 13-s + 16-s − 2·19-s − 25-s + 28-s + 2·31-s − 2·43-s + 49-s − 52-s + 64-s + 2·73-s − 2·76-s − 2·79-s − 91-s − 2·97-s − 100-s + 112-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $1$
Analytic conductor: \(0.408734\)
Root analytic conductor: \(0.639323\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{819} (181, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.251383620\)
\(L(\frac12)\) \(\approx\) \(1.251383620\)
\(L(1)\) 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 \( ( 1 - T )( 1 + T ) \)
5 \( 1 + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 + T )^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( ( 1 - T )^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( 1 + T^{2} \)
43 \( ( 1 + T )^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 - T )^{2} \)
79 \( ( 1 + T )^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( ( 1 + T )^{2} \)
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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

−10.46831915002571710856588775373, −9.857099150156110565114598839409, −8.440098974431187269564506660352, −7.976986090834095275542773452756, −6.95215723222631208877289346109, −6.22411225716002505560485579821, −5.09912208755369812623947972946, −4.15646283745789568143023586820, −2.64628462579650265737685975477, −1.78858881088974432925030621222, 1.78858881088974432925030621222, 2.64628462579650265737685975477, 4.15646283745789568143023586820, 5.09912208755369812623947972946, 6.22411225716002505560485579821, 6.95215723222631208877289346109, 7.976986090834095275542773452756, 8.440098974431187269564506660352, 9.857099150156110565114598839409, 10.46831915002571710856588775373

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