Properties

Label 2-1792-1.1-c1-0-30
Degree $2$
Conductor $1792$
Sign $-1$
Analytic cond. $14.3091$
Root an. cond. $3.78274$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 2·5-s − 7-s + 9-s − 2·13-s − 4·15-s + 6·17-s − 6·19-s + 2·21-s + 4·23-s − 25-s + 4·27-s − 8·31-s − 2·35-s − 8·37-s + 4·39-s + 6·41-s − 8·43-s + 2·45-s + 49-s − 12·51-s + 4·53-s + 12·57-s − 6·59-s + 2·61-s − 63-s − 4·65-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s − 0.554·13-s − 1.03·15-s + 1.45·17-s − 1.37·19-s + 0.436·21-s + 0.834·23-s − 1/5·25-s + 0.769·27-s − 1.43·31-s − 0.338·35-s − 1.31·37-s + 0.640·39-s + 0.937·41-s − 1.21·43-s + 0.298·45-s + 1/7·49-s − 1.68·51-s + 0.549·53-s + 1.58·57-s − 0.781·59-s + 0.256·61-s − 0.125·63-s − 0.496·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-1$
Analytic conductor: \(14.3091\)
Root analytic conductor: \(3.78274\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1792} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1792,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−9.090852512564851802613788286292, −8.073847681736889055930928744294, −6.99246131051166828753464139127, −6.41485741375964366183296639351, −5.47867350522858162884171552310, −5.25891723488831478168816715334, −3.93857515171648208537257789369, −2.73639830864108581497225505140, −1.51014559356040138195013675701, 0, 1.51014559356040138195013675701, 2.73639830864108581497225505140, 3.93857515171648208537257789369, 5.25891723488831478168816715334, 5.47867350522858162884171552310, 6.41485741375964366183296639351, 6.99246131051166828753464139127, 8.073847681736889055930928744294, 9.090852512564851802613788286292

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