Properties

Label 2-1386-1.1-c1-0-4
Degree $2$
Conductor $1386$
Sign $1$
Analytic cond. $11.0672$
Root an. cond. $3.32675$
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 + 4-s + 7-s − 8-s + 11-s + 2·13-s − 14-s + 16-s + 2·19-s − 22-s − 5·25-s − 2·26-s + 28-s + 6·29-s + 2·31-s − 32-s + 2·37-s − 2·38-s − 4·43-s + 44-s + 6·47-s + 49-s + 5·50-s + 2·52-s + 6·53-s − 56-s − 6·58-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 0.301·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.458·19-s − 0.213·22-s − 25-s − 0.392·26-s + 0.188·28-s + 1.11·29-s + 0.359·31-s − 0.176·32-s + 0.328·37-s − 0.324·38-s − 0.609·43-s + 0.150·44-s + 0.875·47-s + 1/7·49-s + 0.707·50-s + 0.277·52-s + 0.824·53-s − 0.133·56-s − 0.787·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1386} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.313716839\)
\(L(\frac12)\) \(\approx\) \(1.313716839\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
11 \( 1 - T \)
good5 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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

−9.575410322452751658082485313148, −8.707141431781925178766899910941, −8.104024357108211588654438142125, −7.27269450626182225527303312385, −6.40050362102582630929450786700, −5.57921544774938402027851071868, −4.44481069307592434914952580537, −3.37789225057720607500704079136, −2.16035280583295696954900337452, −0.960161767014088955334438808530, 0.960161767014088955334438808530, 2.16035280583295696954900337452, 3.37789225057720607500704079136, 4.44481069307592434914952580537, 5.57921544774938402027851071868, 6.40050362102582630929450786700, 7.27269450626182225527303312385, 8.104024357108211588654438142125, 8.707141431781925178766899910941, 9.575410322452751658082485313148

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