↑

Properties

Label 2-2547-283.282-c0-0-4
Degree $2$
Conductor $2547$
Sign $1$
Analytic cond. $1.27111$
Root an. cond. $1.12743$
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·2-s − 1.00·4-s − 1.41i·5-s − 7-s + 2.00·10-s − 11-s + 13-s − 1.41i·14-s − 0.999·16-s − 1.41i·19-s + 1.41i·20-s − 1.41i·22-s + 23-s − 1.00·25-s + 1.41i·26-s + ⋯
L(s)  = 1  + 1.41i·2-s − 1.00·4-s − 1.41i·5-s − 7-s + 2.00·10-s − 11-s + 13-s − 1.41i·14-s − 0.999·16-s − 1.41i·19-s + 1.41i·20-s − 1.41i·22-s + 23-s − 1.00·25-s + 1.41i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2547\)    =    \(3^{2} \cdot 283\)
Sign: $1$
Analytic conductor: \(1.27111\)
Root analytic conductor: \(1.12743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2547} (1414, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2547,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8878379904\)
\(L(\frac12)\) \(\approx\) \(0.8878379904\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
283 \( 1 - T \)
good2 \( 1 - 1.41iT - T^{2} \)
5 \( 1 + 1.41iT - T^{2} \)
7 \( 1 + T + T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 1.41iT - T^{2} \)
23 \( 1 - T + T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 + 1.41iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 + 1.41iT - T^{2} \)
47 \( 1 + 1.41iT - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + T + T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 2T + T^{2} \)
89 \( 1 - T + T^{2} \)
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

−8.813031322149062031736096157151, −8.381786734725076970583318153397, −7.52258329585208277744141463931, −6.72246423115098863736403380844, −6.09044998557728189503409538270, −5.15097376811616061839467123194, −4.86955564260269628770000945605, −3.66083185311367021256757540631, −2.40054227820248313807731092192, −0.60362629309525608176123555523, 1.42175357850838866047713887836, 2.71962470473180361004305161461, 3.14109427540077014989611945302, 3.71205048834642910215329315534, 4.95493129489901418364889277747, 6.26354662231905343526706012635, 6.57307124950493755579005109479, 7.57372210931640267196474545457, 8.498360470112806885640681701212, 9.476172523956910093300225598684

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