# Properties

 Label 2-15e2-45.32-c3-0-27 Degree $2$ Conductor $225$ Sign $-0.413 - 0.910i$ Analytic cond. $13.2754$ Root an. cond. $3.64354$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (5.27 + 1.41i)2-s + (−1.94 + 4.81i)3-s + (18.9 + 10.9i)4-s + (−17.0 + 22.6i)6-s + (−1.13 + 4.22i)7-s + (53.3 + 53.3i)8-s + (−19.4 − 18.7i)9-s + (−5.91 + 3.41i)11-s + (−89.2 + 69.9i)12-s + (6.54 + 24.4i)13-s + (−11.9 + 20.6i)14-s + (118. + 205. i)16-s + (36.0 − 36.0i)17-s + (−76.2 − 126. i)18-s − 59.8i·19-s + ⋯
 L(s)  = 1 + (1.86 + 0.499i)2-s + (−0.373 + 0.927i)3-s + (2.36 + 1.36i)4-s + (−1.16 + 1.54i)6-s + (−0.0610 + 0.227i)7-s + (2.35 + 2.35i)8-s + (−0.720 − 0.693i)9-s + (−0.162 + 0.0936i)11-s + (−2.14 + 1.68i)12-s + (0.139 + 0.521i)13-s + (−0.227 + 0.394i)14-s + (1.85 + 3.21i)16-s + (0.514 − 0.514i)17-s + (−0.998 − 1.65i)18-s − 0.722i·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.413 - 0.910i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$225$$    =    $$3^{2} \cdot 5^{2}$$ Sign: $-0.413 - 0.910i$ Analytic conductor: $$13.2754$$ Root analytic conductor: $$3.64354$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{225} (32, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 225,\ (\ :3/2),\ -0.413 - 0.910i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$2.57345 + 3.99385i$$ $$L(\frac12)$$ $$\approx$$ $$2.57345 + 3.99385i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (1.94 - 4.81i)T$$
5 $$1$$
good2 $$1 + (-5.27 - 1.41i)T + (6.92 + 4i)T^{2}$$
7 $$1 + (1.13 - 4.22i)T + (-297. - 171.5i)T^{2}$$
11 $$1 + (5.91 - 3.41i)T + (665.5 - 1.15e3i)T^{2}$$
13 $$1 + (-6.54 - 24.4i)T + (-1.90e3 + 1.09e3i)T^{2}$$
17 $$1 + (-36.0 + 36.0i)T - 4.91e3iT^{2}$$
19 $$1 + 59.8iT - 6.85e3T^{2}$$
23 $$1 + (50.9 - 13.6i)T + (1.05e4 - 6.08e3i)T^{2}$$
29 $$1 + (-22.1 - 38.2i)T + (-1.21e4 + 2.11e4i)T^{2}$$
31 $$1 + (-67.9 + 117. i)T + (-1.48e4 - 2.57e4i)T^{2}$$
37 $$1 + (233. + 233. i)T + 5.06e4iT^{2}$$
41 $$1 + (-335. - 193. i)T + (3.44e4 + 5.96e4i)T^{2}$$
43 $$1 + (-306. - 82.0i)T + (6.88e4 + 3.97e4i)T^{2}$$
47 $$1 + (316. + 84.9i)T + (8.99e4 + 5.19e4i)T^{2}$$
53 $$1 + (-43.1 - 43.1i)T + 1.48e5iT^{2}$$
59 $$1 + (-248. + 431. i)T + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (-36.3 - 62.8i)T + (-1.13e5 + 1.96e5i)T^{2}$$
67 $$1 + (178. - 47.7i)T + (2.60e5 - 1.50e5i)T^{2}$$
71 $$1 + 563. iT - 3.57e5T^{2}$$
73 $$1 + (84.7 - 84.7i)T - 3.89e5iT^{2}$$
79 $$1 + (-186. + 107. i)T + (2.46e5 - 4.26e5i)T^{2}$$
83 $$1 + (-249. + 930. i)T + (-4.95e5 - 2.85e5i)T^{2}$$
89 $$1 - 288.T + 7.04e5T^{2}$$
97 $$1 + (-425. + 1.58e3i)T + (-7.90e5 - 4.56e5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$