Properties

Label 2-663-1.1-c1-0-5
Degree $2$
Conductor $663$
Sign $1$
Analytic cond. $5.29408$
Root an. cond. $2.30088$
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 − 3-s − 4-s − 4·5-s − 6-s + 2·7-s − 3·8-s + 9-s − 4·10-s + 6·11-s + 12-s − 13-s + 2·14-s + 4·15-s − 16-s + 17-s + 18-s + 4·19-s + 4·20-s − 2·21-s + 6·22-s + 3·24-s + 11·25-s − 26-s − 27-s − 2·28-s − 6·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s − 1.78·5-s − 0.408·6-s + 0.755·7-s − 1.06·8-s + 1/3·9-s − 1.26·10-s + 1.80·11-s + 0.288·12-s − 0.277·13-s + 0.534·14-s + 1.03·15-s − 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s + 0.894·20-s − 0.436·21-s + 1.27·22-s + 0.612·24-s + 11/5·25-s − 0.196·26-s − 0.192·27-s − 0.377·28-s − 1.11·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(663\)    =    \(3 \cdot 13 \cdot 17\)
Sign: $1$
Analytic conductor: \(5.29408\)
Root analytic conductor: \(2.30088\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{663} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 663,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.186139873\)
\(L(\frac12)\) \(\approx\) \(1.186139873\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
13 \( 1 + T \)
17 \( 1 - T \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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

−10.95428777190286288003241299811, −9.556346633123820290554207742334, −8.737638842737171079459728849344, −7.81342569999435272878515329062, −6.98832239970943095443656552142, −5.84386084483901740500080157566, −4.68717336447753969967438730348, −4.18452058198540408772009248966, −3.35760137954518791100219081623, −0.905158356090526027105523059946, 0.905158356090526027105523059946, 3.35760137954518791100219081623, 4.18452058198540408772009248966, 4.68717336447753969967438730348, 5.84386084483901740500080157566, 6.98832239970943095443656552142, 7.81342569999435272878515329062, 8.737638842737171079459728849344, 9.556346633123820290554207742334, 10.95428777190286288003241299811

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