Properties

Label 2-1638-1.1-c1-0-24
Degree $2$
Conductor $1638$
Sign $-1$
Analytic cond. $13.0794$
Root an. cond. $3.61655$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s − 5·11-s − 13-s − 14-s + 16-s + 3·17-s − 19-s + 20-s + 5·22-s − 3·23-s − 4·25-s + 26-s + 28-s − 9·29-s + 4·31-s − 32-s − 3·34-s + 35-s − 11·37-s + 38-s − 40-s − 5·43-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s − 1.50·11-s − 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.229·19-s + 0.223·20-s + 1.06·22-s − 0.625·23-s − 4/5·25-s + 0.196·26-s + 0.188·28-s − 1.67·29-s + 0.718·31-s − 0.176·32-s − 0.514·34-s + 0.169·35-s − 1.80·37-s + 0.162·38-s − 0.158·40-s − 0.762·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-1$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1638} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
7 \( 1 - T \)
13 \( 1 + T \)
good5 \( 1 - T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 15 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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

−9.067467943273849313007184265002, −7.935932481491448532694078819912, −7.80015052507622603521937526703, −6.69232029580076888718309416404, −5.63709318901757604410923700064, −5.14572588019451526117366164450, −3.74734603047492382743471605599, −2.56045395592179615052900667030, −1.71010590596796076771996481178, 0, 1.71010590596796076771996481178, 2.56045395592179615052900667030, 3.74734603047492382743471605599, 5.14572588019451526117366164450, 5.63709318901757604410923700064, 6.69232029580076888718309416404, 7.80015052507622603521937526703, 7.935932481491448532694078819912, 9.067467943273849313007184265002

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