Properties

Label 2-1200-1.1-c1-0-5
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

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·7-s + 9-s − 2·11-s + 6·13-s + 2·17-s − 2·21-s + 4·23-s + 27-s + 8·31-s − 2·33-s + 2·37-s + 6·39-s + 2·41-s + 4·43-s + 8·47-s − 3·49-s + 2·51-s + 6·53-s − 10·59-s + 2·61-s − 2·63-s + 8·67-s + 4·69-s − 12·71-s − 4·73-s + 4·77-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.603·11-s + 1.66·13-s + 0.485·17-s − 0.436·21-s + 0.834·23-s + 0.192·27-s + 1.43·31-s − 0.348·33-s + 0.328·37-s + 0.960·39-s + 0.312·41-s + 0.609·43-s + 1.16·47-s − 3/7·49-s + 0.280·51-s + 0.824·53-s − 1.30·59-s + 0.256·61-s − 0.251·63-s + 0.977·67-s + 0.481·69-s − 1.42·71-s − 0.468·73-s + 0.455·77-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

\(L(1)\) \(\approx\) \(1.990547316\)
\(L(\frac12)\) \(\approx\) \(1.990547316\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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

−9.692635690112158052669440441100, −8.876429119746077603127030187363, −8.236259450125909594320955817325, −7.34830902118093822137916374938, −6.39955394184406631525545593399, −5.66765515312328094809085573246, −4.40421151945420287303938949967, −3.43332661175523197849238647019, −2.66426897593599555100997020524, −1.09380620717066232730427473522, 1.09380620717066232730427473522, 2.66426897593599555100997020524, 3.43332661175523197849238647019, 4.40421151945420287303938949967, 5.66765515312328094809085573246, 6.39955394184406631525545593399, 7.34830902118093822137916374938, 8.236259450125909594320955817325, 8.876429119746077603127030187363, 9.692635690112158052669440441100

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