# Properties

 Label 2-15e2-15.8-c3-0-17 Degree $2$ Conductor $225$ Sign $-0.998 - 0.0618i$ 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 + (0.287 + 0.287i)2-s − 7.83i·4-s + (−15.0 + 15.0i)7-s + (4.55 − 4.55i)8-s − 27.9i·11-s + (−2.49 − 2.49i)13-s − 8.66·14-s − 60.0·16-s + (−67.9 − 67.9i)17-s + 95.2i·19-s + (8.06 − 8.06i)22-s + (−121. + 121. i)23-s − 1.43i·26-s + (117. + 117. i)28-s − 99.0·29-s + ⋯
 L(s)  = 1 + (0.101 + 0.101i)2-s − 0.979i·4-s + (−0.812 + 0.812i)7-s + (0.201 − 0.201i)8-s − 0.767i·11-s + (−0.0533 − 0.0533i)13-s − 0.165·14-s − 0.938·16-s + (−0.968 − 0.968i)17-s + 1.14i·19-s + (0.0781 − 0.0781i)22-s + (−1.10 + 1.10i)23-s − 0.0108i·26-s + (0.796 + 0.796i)28-s − 0.634·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0618i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$225$$    =    $$3^{2} \cdot 5^{2}$$ Sign: $-0.998 - 0.0618i$ Analytic conductor: $$13.2754$$ Root analytic conductor: $$3.64354$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{225} (143, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 225,\ (\ :3/2),\ -0.998 - 0.0618i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.00739727 + 0.238910i$$ $$L(\frac12)$$ $$\approx$$ $$0.00739727 + 0.238910i$$ $$L(\frac{5}{2})$$ 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$$
good2 $$1 + (-0.287 - 0.287i)T + 8iT^{2}$$
7 $$1 + (15.0 - 15.0i)T - 343iT^{2}$$
11 $$1 + 27.9iT - 1.33e3T^{2}$$
13 $$1 + (2.49 + 2.49i)T + 2.19e3iT^{2}$$
17 $$1 + (67.9 + 67.9i)T + 4.91e3iT^{2}$$
19 $$1 - 95.2iT - 6.85e3T^{2}$$
23 $$1 + (121. - 121. i)T - 1.21e4iT^{2}$$
29 $$1 + 99.0T + 2.43e4T^{2}$$
31 $$1 + 28.7T + 2.97e4T^{2}$$
37 $$1 + (271. - 271. i)T - 5.06e4iT^{2}$$
41 $$1 + 453. iT - 6.89e4T^{2}$$
43 $$1 + (30.5 + 30.5i)T + 7.95e4iT^{2}$$
47 $$1 + (254. + 254. i)T + 1.03e5iT^{2}$$
53 $$1 + (-224. + 224. i)T - 1.48e5iT^{2}$$
59 $$1 - 483.T + 2.05e5T^{2}$$
61 $$1 + 264.T + 2.26e5T^{2}$$
67 $$1 + (-498. + 498. i)T - 3.00e5iT^{2}$$
71 $$1 - 609. iT - 3.57e5T^{2}$$
73 $$1 + (-74.6 - 74.6i)T + 3.89e5iT^{2}$$
79 $$1 + 406. iT - 4.93e5T^{2}$$
83 $$1 + (-652. + 652. i)T - 5.71e5iT^{2}$$
89 $$1 + 139.T + 7.04e5T^{2}$$
97 $$1 + (557. - 557. i)T - 9.12e5iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$