# Properties

 Label 2-15e2-25.6-c3-0-17 Degree $2$ Conductor $225$ Sign $0.874 - 0.485i$ 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 + (0.177 + 0.546i)2-s + (6.20 − 4.50i)4-s + (9.45 + 5.96i)5-s − 2.67·7-s + (7.28 + 5.29i)8-s + (−1.58 + 6.22i)10-s + (19.4 + 59.9i)11-s + (4.38 − 13.4i)13-s + (−0.475 − 1.46i)14-s + (17.3 − 53.4i)16-s + (24.5 + 17.8i)17-s + (−22.2 − 16.1i)19-s + (85.5 − 5.57i)20-s + (−29.3 + 21.2i)22-s + (−35.9 − 110. i)23-s + ⋯
 L(s)  = 1 + (0.0627 + 0.193i)2-s + (0.775 − 0.563i)4-s + (0.845 + 0.533i)5-s − 0.144·7-s + (0.321 + 0.233i)8-s + (−0.0500 + 0.196i)10-s + (0.533 + 1.64i)11-s + (0.0934 − 0.287i)13-s + (−0.00907 − 0.0279i)14-s + (0.271 − 0.834i)16-s + (0.350 + 0.254i)17-s + (−0.268 − 0.195i)19-s + (0.956 − 0.0623i)20-s + (−0.283 + 0.206i)22-s + (−0.325 − 1.00i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$225$$    =    $$3^{2} \cdot 5^{2}$$ Sign: $0.874 - 0.485i$ Analytic conductor: $$13.2754$$ Root analytic conductor: $$3.64354$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{225} (181, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 225,\ (\ :3/2),\ 0.874 - 0.485i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$2.52730 + 0.654836i$$ $$L(\frac12)$$ $$\approx$$ $$2.52730 + 0.654836i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
5 $$1 + (-9.45 - 5.96i)T$$
good2 $$1 + (-0.177 - 0.546i)T + (-6.47 + 4.70i)T^{2}$$
7 $$1 + 2.67T + 343T^{2}$$
11 $$1 + (-19.4 - 59.9i)T + (-1.07e3 + 782. i)T^{2}$$
13 $$1 + (-4.38 + 13.4i)T + (-1.77e3 - 1.29e3i)T^{2}$$
17 $$1 + (-24.5 - 17.8i)T + (1.51e3 + 4.67e3i)T^{2}$$
19 $$1 + (22.2 + 16.1i)T + (2.11e3 + 6.52e3i)T^{2}$$
23 $$1 + (35.9 + 110. i)T + (-9.84e3 + 7.15e3i)T^{2}$$
29 $$1 + (32.5 - 23.6i)T + (7.53e3 - 2.31e4i)T^{2}$$
31 $$1 + (-180. - 130. i)T + (9.20e3 + 2.83e4i)T^{2}$$
37 $$1 + (-25.6 + 78.8i)T + (-4.09e4 - 2.97e4i)T^{2}$$
41 $$1 + (44.5 - 137. i)T + (-5.57e4 - 4.05e4i)T^{2}$$
43 $$1 - 433.T + 7.95e4T^{2}$$
47 $$1 + (-371. + 269. i)T + (3.20e4 - 9.87e4i)T^{2}$$
53 $$1 + (200. - 145. i)T + (4.60e4 - 1.41e5i)T^{2}$$
59 $$1 + (95.9 - 295. i)T + (-1.66e5 - 1.20e5i)T^{2}$$
61 $$1 + (-0.644 - 1.98i)T + (-1.83e5 + 1.33e5i)T^{2}$$
67 $$1 + (529. + 384. i)T + (9.29e4 + 2.86e5i)T^{2}$$
71 $$1 + (734. - 533. i)T + (1.10e5 - 3.40e5i)T^{2}$$
73 $$1 + (271. + 835. i)T + (-3.14e5 + 2.28e5i)T^{2}$$
79 $$1 + (-921. + 669. i)T + (1.52e5 - 4.68e5i)T^{2}$$
83 $$1 + (798. + 579. i)T + (1.76e5 + 5.43e5i)T^{2}$$
89 $$1 + (305. + 940. i)T + (-5.70e5 + 4.14e5i)T^{2}$$
97 $$1 + (1.24e3 - 903. i)T + (2.82e5 - 8.68e5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$