Properties

Label 2-8550-1.1-c1-0-135
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 − 2·17-s − 19-s − 2·22-s + 4·23-s − 4·26-s + 2·28-s − 8·31-s + 32-s − 2·34-s − 8·37-s − 38-s + 8·41-s + 6·43-s − 2·44-s + 4·46-s − 12·47-s − 3·49-s − 4·52-s − 6·53-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 − 0.485·17-s − 0.229·19-s − 0.426·22-s + 0.834·23-s − 0.784·26-s + 0.377·28-s − 1.43·31-s + 0.176·32-s − 0.342·34-s − 1.31·37-s − 0.162·38-s + 1.24·41-s + 0.914·43-s − 0.301·44-s + 0.589·46-s − 1.75·47-s − 3/7·49-s − 0.554·52-s − 0.824·53-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 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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

−7.43497398753394446325207128900, −6.77314924840295592221284830831, −5.93309863544358775579937097129, −5.06486408141361635860126041329, −4.87850886269875884523769208147, −3.97790187284038039780780400231, −3.05946013742975761460459534844, −2.31113594193571247930445376114, −1.53081178679857996612768776453, 0, 1.53081178679857996612768776453, 2.31113594193571247930445376114, 3.05946013742975761460459534844, 3.97790187284038039780780400231, 4.87850886269875884523769208147, 5.06486408141361635860126041329, 5.93309863544358775579937097129, 6.77314924840295592221284830831, 7.43497398753394446325207128900

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