# Properties

 Label 2-1200-1.1-c1-0-13 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 $1$

# Learn more

## Dirichlet series

 L(s)  = 1 − 3-s + 9-s − 4·11-s + 2·13-s − 2·17-s + 4·19-s − 8·23-s − 27-s + 6·29-s − 8·31-s + 4·33-s − 6·37-s − 2·39-s − 6·41-s + 4·43-s − 7·49-s + 2·51-s + 2·53-s − 4·57-s − 4·59-s − 2·61-s − 4·67-s + 8·69-s − 8·71-s − 10·73-s + 8·79-s + 81-s + ⋯
 L(s)  = 1 − 0.577·3-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.485·17-s + 0.917·19-s − 1.66·23-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.696·33-s − 0.986·37-s − 0.320·39-s − 0.937·41-s + 0.609·43-s − 49-s + 0.280·51-s + 0.274·53-s − 0.529·57-s − 0.520·59-s − 0.256·61-s − 0.488·67-s + 0.963·69-s − 0.949·71-s − 1.17·73-s + 0.900·79-s + 1/9·81-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: $$1$$ Selberg data: $$(2,\ 1200,\ (\ :1/2),\ -1)$$

## 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$$
3 $$1 + T$$
5 $$1$$
good7 $$1 + p T^{2}$$
11 $$1 + 4 T + p T^{2}$$
13 $$1 - 2 T + p T^{2}$$
17 $$1 + 2 T + p T^{2}$$
19 $$1 - 4 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 + 6 T + p T^{2}$$
41 $$1 + 6 T + p T^{2}$$
43 $$1 - 4 T + p T^{2}$$
47 $$1 + p T^{2}$$
53 $$1 - 2 T + p T^{2}$$
59 $$1 + 4 T + p T^{2}$$
61 $$1 + 2 T + p T^{2}$$
67 $$1 + 4 T + p T^{2}$$
71 $$1 + 8 T + p T^{2}$$
73 $$1 + 10 T + p T^{2}$$
79 $$1 - 8 T + p T^{2}$$
83 $$1 + 4 T + p T^{2}$$
89 $$1 + 6 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

−9.449285690166437837164957292815, −8.428095002482665926481658846183, −7.70640093924588702961218917702, −6.81554110451771320335088635337, −5.85324000516580696107361587496, −5.20634536916068220796571779353, −4.18465656456029023241423917214, −3.04557269583849472425830805016, −1.71961790068640789274174212358, 0, 1.71961790068640789274174212358, 3.04557269583849472425830805016, 4.18465656456029023241423917214, 5.20634536916068220796571779353, 5.85324000516580696107361587496, 6.81554110451771320335088635337, 7.70640093924588702961218917702, 8.428095002482665926481658846183, 9.449285690166437837164957292815