Properties

Label 2-8550-1.1-c1-0-89
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 $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 5·7-s + 8-s + 4·11-s + 13-s + 5·14-s + 16-s − 3·17-s + 19-s + 4·22-s + 7·23-s + 26-s + 5·28-s + 3·29-s − 2·31-s + 32-s − 3·34-s + 2·37-s + 38-s + 6·41-s − 6·43-s + 4·44-s + 7·46-s + 18·49-s + 52-s − 13·53-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 1.88·7-s + 0.353·8-s + 1.20·11-s + 0.277·13-s + 1.33·14-s + 1/4·16-s − 0.727·17-s + 0.229·19-s + 0.852·22-s + 1.45·23-s + 0.196·26-s + 0.944·28-s + 0.557·29-s − 0.359·31-s + 0.176·32-s − 0.514·34-s + 0.328·37-s + 0.162·38-s + 0.937·41-s − 0.914·43-s + 0.603·44-s + 1.03·46-s + 18/7·49-s + 0.138·52-s − 1.78·53-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: \(0\)
Selberg data: \((2,\ 8550,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.136091102\)
\(L(\frac12)\) \(\approx\) \(5.136091102\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 13 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−7.64412675368342072100402643347, −7.07704541117290869170938897697, −6.35430569173332358701577738897, −5.60181065616457415736945006525, −4.73764288830413259853760333013, −4.52726401175650390589229310145, −3.63706267746930921054569078234, −2.68059787783471348034578632155, −1.68490358819888880085777025199, −1.14285782158350026604192915140, 1.14285782158350026604192915140, 1.68490358819888880085777025199, 2.68059787783471348034578632155, 3.63706267746930921054569078234, 4.52726401175650390589229310145, 4.73764288830413259853760333013, 5.60181065616457415736945006525, 6.35430569173332358701577738897, 7.07704541117290869170938897697, 7.64412675368342072100402643347

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