Properties

Label 2-1386-1.1-c1-0-10
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 + 4·17-s + 6·19-s + 22-s + 4·23-s − 5·25-s − 2·26-s − 28-s + 10·29-s + 6·31-s + 32-s + 4·34-s − 6·37-s + 6·38-s + 12·41-s − 8·43-s + 44-s + 4·46-s − 2·47-s + 49-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 + 0.970·17-s + 1.37·19-s + 0.213·22-s + 0.834·23-s − 25-s − 0.392·26-s − 0.188·28-s + 1.85·29-s + 1.07·31-s + 0.176·32-s + 0.685·34-s − 0.986·37-s + 0.973·38-s + 1.87·41-s − 1.21·43-s + 0.150·44-s + 0.589·46-s − 0.291·47-s + 1/7·49-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.685417362\)
\(L(\frac12)\) \(\approx\) \(2.685417362\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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

−9.891429834212501423422030953706, −8.751364050919664408948964913092, −7.76644875192378882330336596739, −7.06243874841004507589844210062, −6.19912680921072839463289428224, −5.34458459387631487464034223256, −4.55153014528148769824162116427, −3.42266397038887650714886536973, −2.70873132071359508119536298434, −1.15718775335031232118843306899, 1.15718775335031232118843306899, 2.70873132071359508119536298434, 3.42266397038887650714886536973, 4.55153014528148769824162116427, 5.34458459387631487464034223256, 6.19912680921072839463289428224, 7.06243874841004507589844210062, 7.76644875192378882330336596739, 8.751364050919664408948964913092, 9.891429834212501423422030953706

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