↑

Properties

Label 2-2284-571.570-c0-0-0
Degree $2$
Conductor $2284$
Sign $-1$
Analytic cond. $1.13986$
Root an. cond. $1.06764$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·3-s + 1.41i·7-s − 1.00·9-s − 11-s − 13-s − 1.41i·17-s + 1.41i·19-s − 2.00·21-s − 25-s + 29-s − 31-s − 1.41i·33-s + 37-s − 1.41i·39-s − 43-s + ⋯
L(s)  = 1  + 1.41i·3-s + 1.41i·7-s − 1.00·9-s − 11-s − 13-s − 1.41i·17-s + 1.41i·19-s − 2.00·21-s − 25-s + 29-s − 31-s − 1.41i·33-s + 37-s − 1.41i·39-s − 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2284 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2284 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2284\)    =    \(2^{2} \cdot 571\)
Sign: $-1$
Analytic conductor: \(1.13986\)
Root analytic conductor: \(1.06764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2284} (1141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2284,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7891027230\)
\(L(\frac12)\) \(\approx\) \(0.7891027230\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
571 \( 1 + T \)
good3 \( 1 - 1.41iT - T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 - 1.41iT - T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + T + T^{2} \)
17 \( 1 + 1.41iT - T^{2} \)
19 \( 1 - 1.41iT - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 1.41iT - T^{2} \)
59 \( 1 - T + T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T + T^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 + T + T^{2} \)
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

−9.626696576136582098559407653750, −9.075875008255986097068368578784, −8.163748291815302308760251300345, −7.48614830059550937728706932221, −6.19395937383131818480816681137, −5.21137449063717361377527031354, −5.13886584721282030330492849050, −3.98204993820424709002678325344, −2.94716104047900023683264071263, −2.26139699709297765844543382653, 0.50786436677373899078888570264, 1.78958918401679965108787383469, 2.68404818616556325221722776429, 3.91572666739582419625029225793, 4.84936951471410830189494931530, 5.84335997026537595159833186081, 6.90575860835752836702358534013, 7.09838974145605233637928611250, 7.968587237526381360633218482412, 8.379539088629211489473166058808

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