# Properties

 Label 2-15e2-45.34-c3-0-17 Degree $2$ Conductor $225$ Sign $0.125 + 0.992i$ 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 + (−3.96 − 2.28i)2-s + (−3.96 + 3.36i)3-s + (6.45 + 11.1i)4-s + (23.3 − 4.26i)6-s + (−17.4 − 10.0i)7-s − 22.4i·8-s + (4.37 − 26.6i)9-s + (−33.1 + 57.4i)11-s + (−63.2 − 22.5i)12-s + (−40.5 + 23.4i)13-s + (45.9 + 79.6i)14-s + (0.237 − 0.411i)16-s + 47.6i·17-s + (−78.2 + 95.5i)18-s + 9.95·19-s + ⋯
 L(s)  = 1 + (−1.40 − 0.808i)2-s + (−0.762 + 0.647i)3-s + (0.807 + 1.39i)4-s + (1.59 − 0.290i)6-s + (−0.940 − 0.543i)7-s − 0.993i·8-s + (0.162 − 0.986i)9-s + (−0.909 + 1.57i)11-s + (−1.52 − 0.543i)12-s + (−0.864 + 0.499i)13-s + (0.878 + 1.52i)14-s + (0.00371 − 0.00643i)16-s + 0.679i·17-s + (−1.02 + 1.25i)18-s + 0.120·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.125 + 0.992i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.125 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$225$$    =    $$3^{2} \cdot 5^{2}$$ Sign: $0.125 + 0.992i$ Analytic conductor: $$13.2754$$ Root analytic conductor: $$3.64354$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{225} (124, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 225,\ (\ :3/2),\ 0.125 + 0.992i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.162177 - 0.142886i$$ $$L(\frac12)$$ $$\approx$$ $$0.162177 - 0.142886i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (3.96 - 3.36i)T$$
5 $$1$$
good2 $$1 + (3.96 + 2.28i)T + (4 + 6.92i)T^{2}$$
7 $$1 + (17.4 + 10.0i)T + (171.5 + 297. i)T^{2}$$
11 $$1 + (33.1 - 57.4i)T + (-665.5 - 1.15e3i)T^{2}$$
13 $$1 + (40.5 - 23.4i)T + (1.09e3 - 1.90e3i)T^{2}$$
17 $$1 - 47.6iT - 4.91e3T^{2}$$
19 $$1 - 9.95T + 6.85e3T^{2}$$
23 $$1 + (-8.30 + 4.79i)T + (6.08e3 - 1.05e4i)T^{2}$$
29 $$1 + (89.3 - 154. i)T + (-1.21e4 - 2.11e4i)T^{2}$$
31 $$1 + (77.0 + 133. i)T + (-1.48e4 + 2.57e4i)T^{2}$$
37 $$1 + 248. iT - 5.06e4T^{2}$$
41 $$1 + (124. + 216. i)T + (-3.44e4 + 5.96e4i)T^{2}$$
43 $$1 + (-183. - 106. i)T + (3.97e4 + 6.88e4i)T^{2}$$
47 $$1 + (-411. - 237. i)T + (5.19e4 + 8.99e4i)T^{2}$$
53 $$1 + 546. iT - 1.48e5T^{2}$$
59 $$1 + (209. + 363. i)T + (-1.02e5 + 1.77e5i)T^{2}$$
61 $$1 + (-272. + 472. i)T + (-1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (-387. + 223. i)T + (1.50e5 - 2.60e5i)T^{2}$$
71 $$1 - 409.T + 3.57e5T^{2}$$
73 $$1 + 358. iT - 3.89e5T^{2}$$
79 $$1 + (325. - 564. i)T + (-2.46e5 - 4.26e5i)T^{2}$$
83 $$1 + (-704. - 406. i)T + (2.85e5 + 4.95e5i)T^{2}$$
89 $$1 - 200.T + 7.04e5T^{2}$$
97 $$1 + (-218. - 126. i)T + (4.56e5 + 7.90e5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$