↑

Properties

Label 2-53550-1.1-c1-0-51
Degree $2$
Conductor $53550$
Sign $-1$
Analytic cond. $427.598$
Root an. cond. $20.6784$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 7-s − 8-s − 2·11-s + 2·13-s + 14-s + 16-s + 17-s − 4·19-s + 2·22-s + 2·23-s − 2·26-s − 28-s − 2·29-s − 8·31-s − 32-s − 34-s − 6·37-s + 4·38-s + 10·41-s − 2·44-s − 2·46-s + 2·47-s + 49-s + 2·52-s + 10·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 0.603·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.917·19-s + 0.426·22-s + 0.417·23-s − 0.392·26-s − 0.188·28-s − 0.371·29-s − 1.43·31-s − 0.176·32-s − 0.171·34-s − 0.986·37-s + 0.648·38-s + 1.56·41-s − 0.301·44-s − 0.294·46-s + 0.291·47-s + 1/7·49-s + 0.277·52-s + 1.37·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(53550\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 17\)
Sign: $-1$
Analytic conductor: \(427.598\)
Root analytic conductor: \(20.6784\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 53550,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
17 \( 1 - T \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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

−14.74602057896878, −14.32818928977529, −13.58734390304983, −13.03518725104678, −12.73350954938077, −12.15996000276360, −11.45105991614865, −11.04277181711481, −10.40338633756543, −10.27434066579860, −9.412117681464382, −8.896231350253367, −8.671617581257774, −7.845828110067799, −7.402509326803753, −6.934774109228152, −6.197299880457222, −5.744065971071630, −5.192544599704474, −4.304312705331045, −3.711627521886727, −3.060870026578368, −2.342106291683861, −1.734802576543296, −0.8187676830112184, 0, 0.8187676830112184, 1.734802576543296, 2.342106291683861, 3.060870026578368, 3.711627521886727, 4.304312705331045, 5.192544599704474, 5.744065971071630, 6.197299880457222, 6.934774109228152, 7.402509326803753, 7.845828110067799, 8.671617581257774, 8.896231350253367, 9.412117681464382, 10.27434066579860, 10.40338633756543, 11.04277181711481, 11.45105991614865, 12.15996000276360, 12.73350954938077, 13.03518725104678, 13.58734390304983, 14.32818928977529, 14.74602057896878

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