Properties

Label 2-2842-1.1-c1-0-8
Degree $2$
Conductor $2842$
Sign $1$
Analytic cond. $22.6934$
Root an. cond. $4.76376$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 4-s − 2·5-s + 2·6-s − 8-s + 9-s + 2·10-s + 4·11-s − 2·12-s + 2·13-s + 4·15-s + 16-s + 4·17-s − 18-s − 2·19-s − 2·20-s − 4·22-s + 2·24-s − 25-s − 2·26-s + 4·27-s − 29-s − 4·30-s + 2·31-s − 32-s − 8·33-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.894·5-s + 0.816·6-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 1.20·11-s − 0.577·12-s + 0.554·13-s + 1.03·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s − 0.458·19-s − 0.447·20-s − 0.852·22-s + 0.408·24-s − 1/5·25-s − 0.392·26-s + 0.769·27-s − 0.185·29-s − 0.730·30-s + 0.359·31-s − 0.176·32-s − 1.39·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2842\)    =    \(2 \cdot 7^{2} \cdot 29\)
Sign: $1$
Analytic conductor: \(22.6934\)
Root analytic conductor: \(4.76376\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2842,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6111056509\)
\(L(\frac12)\) \(\approx\) \(0.6111056509\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
7 \( 1 \)
29 \( 1 + T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.605789181209095450571393447634, −8.181156406370071373473225265699, −7.19077389091838852362366810243, −6.54334625454827283506383172144, −5.92954536041548618947010825248, −5.00946084477045769784252885528, −4.00056290715485073203301368850, −3.24313368198724016020456507067, −1.61885400056636754162386078150, −0.59303062997997118440756794829, 0.59303062997997118440756794829, 1.61885400056636754162386078150, 3.24313368198724016020456507067, 4.00056290715485073203301368850, 5.00946084477045769784252885528, 5.92954536041548618947010825248, 6.54334625454827283506383172144, 7.19077389091838852362366810243, 8.181156406370071373473225265699, 8.605789181209095450571393447634

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