# Properties

 Label 2-300-300.59-c1-0-48 Degree $2$ Conductor $300$ Sign $-0.153 + 0.988i$ 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 + (0.723 − 1.21i)2-s + (1.73 − 0.0711i)3-s + (−0.954 − 1.75i)4-s + (−1.84 − 1.25i)5-s + (1.16 − 2.15i)6-s + 2.80·7-s + (−2.82 − 0.111i)8-s + (2.98 − 0.246i)9-s + (−2.86 + 1.33i)10-s + (−1.50 − 1.08i)11-s + (−1.77 − 2.97i)12-s + (2.01 + 2.76i)13-s + (2.02 − 3.40i)14-s + (−3.28 − 2.04i)15-s + (−2.17 + 3.35i)16-s + (0.345 − 1.06i)17-s + ⋯
 L(s)  = 1 + (0.511 − 0.859i)2-s + (0.999 − 0.0410i)3-s + (−0.477 − 0.878i)4-s + (−0.826 − 0.562i)5-s + (0.475 − 0.879i)6-s + 1.06·7-s + (−0.999 − 0.0393i)8-s + (0.996 − 0.0820i)9-s + (−0.906 + 0.422i)10-s + (−0.452 − 0.328i)11-s + (−0.512 − 0.858i)12-s + (0.558 + 0.768i)13-s + (0.542 − 0.910i)14-s + (−0.849 − 0.528i)15-s + (−0.544 + 0.838i)16-s + (0.0837 − 0.257i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.153 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.153 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$300$$    =    $$2^{2} \cdot 3 \cdot 5^{2}$$ Sign: $-0.153 + 0.988i$ Analytic conductor: $$2.39551$$ Root analytic conductor: $$1.54774$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{300} (59, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 300,\ (\ :1/2),\ -0.153 + 0.988i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.31786 - 1.53814i$$ $$L(\frac12)$$ $$\approx$$ $$1.31786 - 1.53814i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.723 + 1.21i)T$$
3 $$1 + (-1.73 + 0.0711i)T$$
5 $$1 + (1.84 + 1.25i)T$$
good7 $$1 - 2.80T + 7T^{2}$$
11 $$1 + (1.50 + 1.08i)T + (3.39 + 10.4i)T^{2}$$
13 $$1 + (-2.01 - 2.76i)T + (-4.01 + 12.3i)T^{2}$$
17 $$1 + (-0.345 + 1.06i)T + (-13.7 - 9.99i)T^{2}$$
19 $$1 + (7.11 + 2.31i)T + (15.3 + 11.1i)T^{2}$$
23 $$1 + (1.23 - 1.70i)T + (-7.10 - 21.8i)T^{2}$$
29 $$1 + (-3.56 + 1.15i)T + (23.4 - 17.0i)T^{2}$$
31 $$1 + (-8.53 - 2.77i)T + (25.0 + 18.2i)T^{2}$$
37 $$1 + (-4.97 - 6.84i)T + (-11.4 + 35.1i)T^{2}$$
41 $$1 + (-3.81 - 5.25i)T + (-12.6 + 38.9i)T^{2}$$
43 $$1 + 7.87T + 43T^{2}$$
47 $$1 + (0.999 - 0.324i)T + (38.0 - 27.6i)T^{2}$$
53 $$1 + (0.595 + 1.83i)T + (-42.8 + 31.1i)T^{2}$$
59 $$1 + (6.91 - 5.02i)T + (18.2 - 56.1i)T^{2}$$
61 $$1 + (0.730 + 0.530i)T + (18.8 + 58.0i)T^{2}$$
67 $$1 + (-0.462 + 1.42i)T + (-54.2 - 39.3i)T^{2}$$
71 $$1 + (1.11 + 3.41i)T + (-57.4 + 41.7i)T^{2}$$
73 $$1 + (-7.45 + 10.2i)T + (-22.5 - 69.4i)T^{2}$$
79 $$1 + (4.76 - 1.54i)T + (63.9 - 46.4i)T^{2}$$
83 $$1 + (5.84 + 1.89i)T + (67.1 + 48.7i)T^{2}$$
89 $$1 + (9.96 - 13.7i)T + (-27.5 - 84.6i)T^{2}$$
97 $$1 + (16.7 - 5.45i)T + (78.4 - 57.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$