Properties

Label 2-1200-1.1-c1-0-7
Degree $2$
Conductor $1200$
Sign $1$
Analytic cond. $9.58204$
Root an. cond. $3.09548$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s + 4·11-s − 6·13-s + 6·17-s + 4·19-s + 27-s − 2·29-s + 8·31-s + 4·33-s + 2·37-s − 6·39-s − 6·41-s + 12·43-s + 8·47-s − 7·49-s + 6·51-s − 6·53-s + 4·57-s − 12·59-s + 14·61-s + 4·67-s − 8·71-s + 6·73-s + 8·79-s + 81-s − 12·83-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s + 1.20·11-s − 1.66·13-s + 1.45·17-s + 0.917·19-s + 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.696·33-s + 0.328·37-s − 0.960·39-s − 0.937·41-s + 1.82·43-s + 1.16·47-s − 49-s + 0.840·51-s − 0.824·53-s + 0.529·57-s − 1.56·59-s + 1.79·61-s + 0.488·67-s − 0.949·71-s + 0.702·73-s + 0.900·79-s + 1/9·81-s − 1.31·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(9.58204\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.158154419\)
\(L(\frac12)\) \(\approx\) \(2.158154419\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−9.656997109338605303594754183316, −9.132292746390262782909630572480, −7.960956096935259033265277502764, −7.45213305414427898234403856335, −6.54805783058264381139760383347, −5.45085131568784486697698480921, −4.51593385550500718030658955038, −3.48641614075377419176346641099, −2.54593173740013447270825393795, −1.16675507013472100447724827678, 1.16675507013472100447724827678, 2.54593173740013447270825393795, 3.48641614075377419176346641099, 4.51593385550500718030658955038, 5.45085131568784486697698480921, 6.54805783058264381139760383347, 7.45213305414427898234403856335, 7.960956096935259033265277502764, 9.132292746390262782909630572480, 9.656997109338605303594754183316

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