# Properties

 Label 2-1200-15.2-c1-0-33 Degree $2$ Conductor $1200$ Sign $-0.973 + 0.229i$ Analytic cond. $9.58204$ Root an. cond. $3.09548$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.22 − 1.22i)3-s + (−2.44 − 2.44i)7-s − 2.99i·9-s + (−4.89 + 4.89i)13-s − 8i·19-s − 5.99·21-s + (−3.67 − 3.67i)27-s − 4·31-s + (−4.89 − 4.89i)37-s + 11.9i·39-s + (−7.34 + 7.34i)43-s + 4.99i·49-s + (−9.79 − 9.79i)57-s + 14·61-s + (−7.34 + 7.34i)63-s + ⋯
 L(s)  = 1 + (0.707 − 0.707i)3-s + (−0.925 − 0.925i)7-s − 0.999i·9-s + (−1.35 + 1.35i)13-s − 1.83i·19-s − 1.30·21-s + (−0.707 − 0.707i)27-s − 0.718·31-s + (−0.805 − 0.805i)37-s + 1.92i·39-s + (−1.12 + 1.12i)43-s + 0.714i·49-s + (−1.29 − 1.29i)57-s + 1.79·61-s + (−0.925 + 0.925i)63-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1200$$    =    $$2^{4} \cdot 3 \cdot 5^{2}$$ Sign: $-0.973 + 0.229i$ Analytic conductor: $$9.58204$$ Root analytic conductor: $$3.09548$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1200} (257, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1200,\ (\ :1/2),\ -0.973 + 0.229i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.9677605900$$ $$L(\frac12)$$ $$\approx$$ $$0.9677605900$$ $$L(\frac{3}{2})$$ 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 + (-1.22 + 1.22i)T$$
5 $$1$$
good7 $$1 + (2.44 + 2.44i)T + 7iT^{2}$$
11 $$1 - 11T^{2}$$
13 $$1 + (4.89 - 4.89i)T - 13iT^{2}$$
17 $$1 - 17iT^{2}$$
19 $$1 + 8iT - 19T^{2}$$
23 $$1 + 23iT^{2}$$
29 $$1 + 29T^{2}$$
31 $$1 + 4T + 31T^{2}$$
37 $$1 + (4.89 + 4.89i)T + 37iT^{2}$$
41 $$1 - 41T^{2}$$
43 $$1 + (7.34 - 7.34i)T - 43iT^{2}$$
47 $$1 - 47iT^{2}$$
53 $$1 + 53iT^{2}$$
59 $$1 + 59T^{2}$$
61 $$1 - 14T + 61T^{2}$$
67 $$1 + (2.44 + 2.44i)T + 67iT^{2}$$
71 $$1 - 71T^{2}$$
73 $$1 + (-9.79 + 9.79i)T - 73iT^{2}$$
79 $$1 + 4iT - 79T^{2}$$
83 $$1 + 83iT^{2}$$
89 $$1 + 89T^{2}$$
97 $$1 + (9.79 + 9.79i)T + 97iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$