Properties

Label 2-799-1.1-c1-0-24
Degree $2$
Conductor $799$
Sign $1$
Analytic cond. $6.38004$
Root an. cond. $2.52587$
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-s + 2·3-s − 4-s + 4·5-s − 2·6-s − 2·7-s + 3·8-s + 9-s − 4·10-s − 2·12-s + 2·13-s + 2·14-s + 8·15-s − 16-s + 17-s − 18-s + 4·19-s − 4·20-s − 4·21-s − 4·23-s + 6·24-s + 11·25-s − 2·26-s − 4·27-s + 2·28-s + 8·29-s − 8·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s − 1/2·4-s + 1.78·5-s − 0.816·6-s − 0.755·7-s + 1.06·8-s + 1/3·9-s − 1.26·10-s − 0.577·12-s + 0.554·13-s + 0.534·14-s + 2.06·15-s − 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.917·19-s − 0.894·20-s − 0.872·21-s − 0.834·23-s + 1.22·24-s + 11/5·25-s − 0.392·26-s − 0.769·27-s + 0.377·28-s + 1.48·29-s − 1.46·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(799\)    =    \(17 \cdot 47\)
Sign: $1$
Analytic conductor: \(6.38004\)
Root analytic conductor: \(2.52587\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 799,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.782342728\)
\(L(\frac12)\) \(\approx\) \(1.782342728\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 - T \)
47 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 - 2 T + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−9.923805233114826705426209355465, −9.510500167185474756907047741981, −8.659648938238949671437076516991, −8.185517248538253094842685346692, −6.87454595313213151825457041697, −5.97785207334378606624688271601, −4.95558297373443957549295237784, −3.49977739762465808068312303580, −2.51032747149775727153897197062, −1.32013274406742367289000975025, 1.32013274406742367289000975025, 2.51032747149775727153897197062, 3.49977739762465808068312303580, 4.95558297373443957549295237784, 5.97785207334378606624688271601, 6.87454595313213151825457041697, 8.185517248538253094842685346692, 8.659648938238949671437076516991, 9.510500167185474756907047741981, 9.923805233114826705426209355465

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