Properties

Label 2-8550-1.1-c1-0-141
Degree $2$
Conductor $8550$
Sign $-1$
Analytic cond. $68.2720$
Root an. cond. $8.26269$
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 + 2·7-s + 8-s − 2·11-s + 4·13-s + 2·14-s + 16-s − 6·17-s + 19-s − 2·22-s − 8·23-s + 4·26-s + 2·28-s − 6·29-s − 8·31-s + 32-s − 6·34-s + 8·37-s + 38-s − 12·41-s − 2·44-s − 8·46-s − 3·49-s + 4·52-s − 10·53-s + 2·56-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.755·7-s + 0.353·8-s − 0.603·11-s + 1.10·13-s + 0.534·14-s + 1/4·16-s − 1.45·17-s + 0.229·19-s − 0.426·22-s − 1.66·23-s + 0.784·26-s + 0.377·28-s − 1.11·29-s − 1.43·31-s + 0.176·32-s − 1.02·34-s + 1.31·37-s + 0.162·38-s − 1.87·41-s − 0.301·44-s − 1.17·46-s − 3/7·49-s + 0.554·52-s − 1.37·53-s + 0.267·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8550\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(68.2720\)
Root analytic conductor: \(8.26269\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8550,\ (\ :1/2),\ -1)\)

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 \( 1 \)
19 \( 1 - T \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
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

−7.55196063058328307116631211251, −6.50720297645715505233428197854, −6.06836678665006477970710088974, −5.28754380955079178244411795767, −4.62939916428607101352862148388, −3.93555935553654627582589151672, −3.24941784356345959590610351224, −2.09698326370647580425927672294, −1.64728875536469951271641601528, 0, 1.64728875536469951271641601528, 2.09698326370647580425927672294, 3.24941784356345959590610351224, 3.93555935553654627582589151672, 4.62939916428607101352862148388, 5.28754380955079178244411795767, 6.06836678665006477970710088974, 6.50720297645715505233428197854, 7.55196063058328307116631211251

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