# Properties

 Label 2-1200-15.14-c2-0-56 Degree $2$ Conductor $1200$ Sign $-0.447 + 0.894i$ Analytic cond. $32.6976$ Root an. cond. $5.71818$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 3i·3-s − 13i·7-s − 9·9-s + 23i·13-s + 11·19-s + 39·21-s − 27i·27-s − 59·31-s − 26i·37-s − 69·39-s − 83i·43-s − 120·49-s + 33i·57-s − 121·61-s + 117i·63-s + ⋯
 L(s)  = 1 + i·3-s − 1.85i·7-s − 9-s + 1.76i·13-s + 0.578·19-s + 1.85·21-s − i·27-s − 1.90·31-s − 0.702i·37-s − 1.76·39-s − 1.93i·43-s − 2.44·49-s + 0.578i·57-s − 1.98·61-s + 1.85i·63-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1200$$    =    $$2^{4} \cdot 3 \cdot 5^{2}$$ Sign: $-0.447 + 0.894i$ Analytic conductor: $$32.6976$$ Root analytic conductor: $$5.71818$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{1200} (449, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1200,\ (\ :1),\ -0.447 + 0.894i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.5414333018$$ $$L(\frac12)$$ $$\approx$$ $$0.5414333018$$ $$L(2)$$ 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 - 3iT$$
5 $$1$$
good7 $$1 + 13iT - 49T^{2}$$
11 $$1 - 121T^{2}$$
13 $$1 - 23iT - 169T^{2}$$
17 $$1 + 289T^{2}$$
19 $$1 - 11T + 361T^{2}$$
23 $$1 + 529T^{2}$$
29 $$1 - 841T^{2}$$
31 $$1 + 59T + 961T^{2}$$
37 $$1 + 26iT - 1.36e3T^{2}$$
41 $$1 - 1.68e3T^{2}$$
43 $$1 + 83iT - 1.84e3T^{2}$$
47 $$1 + 2.20e3T^{2}$$
53 $$1 + 2.80e3T^{2}$$
59 $$1 - 3.48e3T^{2}$$
61 $$1 + 121T + 3.72e3T^{2}$$
67 $$1 + 13iT - 4.48e3T^{2}$$
71 $$1 - 5.04e3T^{2}$$
73 $$1 + 46iT - 5.32e3T^{2}$$
79 $$1 + 142T + 6.24e3T^{2}$$
83 $$1 + 6.88e3T^{2}$$
89 $$1 - 7.92e3T^{2}$$
97 $$1 + 167iT - 9.40e3T^{2}$$
show less
$$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

−9.389777395671716201513469034411, −8.728600786314345023625280564125, −7.44603916369347235577420959500, −7.00816169936187185627896678697, −5.83606009752520577561577590139, −4.69938630315114602413932672281, −4.07092034965117253693955869920, −3.41512080014130764067236388147, −1.74377695506693417430781020040, −0.15677652851306412910321358092, 1.42332371151406899348358766287, 2.62016294013058508604226690154, 3.19981813948027966126231581276, 5.13650467608733987479659882388, 5.66828971538788621540513019141, 6.32804281240571736527096397442, 7.54488443915516510283392574198, 8.114061847201436409623374491730, 8.881024066759561667128452120861, 9.596125079566744108661686525046