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Properties

Label 2-832-52.51-c0-0-0
Degree $2$
Conductor $832$
Sign $1$
Analytic cond. $0.415222$
Root an. cond. $0.644377$
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  + 9-s + 13-s − 2·17-s + 25-s + 2·29-s − 49-s − 2·53-s − 2·61-s + 81-s − 2·101-s − 2·113-s + 117-s + ⋯
L(s)  = 1  + 9-s + 13-s − 2·17-s + 25-s + 2·29-s − 49-s − 2·53-s − 2·61-s + 81-s − 2·101-s − 2·113-s + 117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(832\)    =    \(2^{6} \cdot 13\)
Sign: $1$
Analytic conductor: \(0.415222\)
Root analytic conductor: \(0.644377\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{832} (831, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 832,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.054208564\)
\(L(\frac12)\) \(\approx\) \(1.054208564\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−10.61272988492612035472853133947, −9.511338464276738053917750322625, −8.762092450649959004989646937029, −7.958285701860560209257731672154, −6.71609034750672152948793715800, −6.40641510857905777119226283577, −4.85192577951185885970913869999, −4.23360727930307929297987245171, −2.91572923999098473238753973839, −1.49651234364975275124565400998, 1.49651234364975275124565400998, 2.91572923999098473238753973839, 4.23360727930307929297987245171, 4.85192577951185885970913869999, 6.40641510857905777119226283577, 6.71609034750672152948793715800, 7.958285701860560209257731672154, 8.762092450649959004989646937029, 9.511338464276738053917750322625, 10.61272988492612035472853133947

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