# Properties

 Label 2-300-300.71-c1-0-6 Degree $2$ Conductor $300$ Sign $0.850 - 0.525i$ Analytic cond. $2.39551$ Root an. cond. $1.54774$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.936 − 1.05i)2-s + (1.00 − 1.40i)3-s + (−0.246 + 1.98i)4-s + (−1.49 + 1.66i)5-s + (−2.43 + 0.250i)6-s + 4.44i·7-s + (2.33 − 1.59i)8-s + (−0.967 − 2.83i)9-s + (3.16 + 0.0306i)10-s + (−1.38 + 4.26i)11-s + (2.54 + 2.34i)12-s + (1.09 + 3.38i)13-s + (4.71 − 4.16i)14-s + (0.831 + 3.78i)15-s + (−3.87 − 0.979i)16-s + (0.229 + 0.316i)17-s + ⋯
 L(s)  = 1 + (−0.662 − 0.749i)2-s + (0.582 − 0.813i)3-s + (−0.123 + 0.992i)4-s + (−0.669 + 0.743i)5-s + (−0.994 + 0.102i)6-s + 1.68i·7-s + (0.825 − 0.564i)8-s + (−0.322 − 0.946i)9-s + (0.999 + 0.00969i)10-s + (−0.417 + 1.28i)11-s + (0.735 + 0.677i)12-s + (0.304 + 0.938i)13-s + (1.26 − 1.11i)14-s + (0.214 + 0.976i)15-s + (−0.969 − 0.244i)16-s + (0.0557 + 0.0767i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$300$$    =    $$2^{2} \cdot 3 \cdot 5^{2}$$ Sign: $0.850 - 0.525i$ Analytic conductor: $$2.39551$$ Root analytic conductor: $$1.54774$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{300} (71, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 300,\ (\ :1/2),\ 0.850 - 0.525i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.812441 + 0.230593i$$ $$L(\frac12)$$ $$\approx$$ $$0.812441 + 0.230593i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.936 + 1.05i)T$$
3 $$1 + (-1.00 + 1.40i)T$$
5 $$1 + (1.49 - 1.66i)T$$
good7 $$1 - 4.44iT - 7T^{2}$$
11 $$1 + (1.38 - 4.26i)T + (-8.89 - 6.46i)T^{2}$$
13 $$1 + (-1.09 - 3.38i)T + (-10.5 + 7.64i)T^{2}$$
17 $$1 + (-0.229 - 0.316i)T + (-5.25 + 16.1i)T^{2}$$
19 $$1 + (1.18 + 1.63i)T + (-5.87 + 18.0i)T^{2}$$
23 $$1 + (0.927 - 2.85i)T + (-18.6 - 13.5i)T^{2}$$
29 $$1 + (-4.65 + 6.40i)T + (-8.96 - 27.5i)T^{2}$$
31 $$1 + (0.713 + 0.981i)T + (-9.57 + 29.4i)T^{2}$$
37 $$1 + (0.145 + 0.447i)T + (-29.9 + 21.7i)T^{2}$$
41 $$1 + (-7.88 + 2.56i)T + (33.1 - 24.0i)T^{2}$$
43 $$1 - 5.44iT - 43T^{2}$$
47 $$1 + (-5.11 - 3.71i)T + (14.5 + 44.6i)T^{2}$$
53 $$1 + (1.31 - 1.81i)T + (-16.3 - 50.4i)T^{2}$$
59 $$1 + (-0.692 - 2.13i)T + (-47.7 + 34.6i)T^{2}$$
61 $$1 + (1.80 - 5.55i)T + (-49.3 - 35.8i)T^{2}$$
67 $$1 + (-4.83 - 6.65i)T + (-20.7 + 63.7i)T^{2}$$
71 $$1 + (-1.00 - 0.727i)T + (21.9 + 67.5i)T^{2}$$
73 $$1 + (-4.53 + 13.9i)T + (-59.0 - 42.9i)T^{2}$$
79 $$1 + (-5.87 + 8.07i)T + (-24.4 - 75.1i)T^{2}$$
83 $$1 + (4.74 - 3.44i)T + (25.6 - 78.9i)T^{2}$$
89 $$1 + (13.5 + 4.39i)T + (72.0 + 52.3i)T^{2}$$
97 $$1 + (-9.45 - 6.86i)T + (29.9 + 92.2i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$