# Properties

 Degree 2 Conductor $3 \cdot 5$ Sign $0.846 - 0.533i$ Motivic weight 3 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (2.66 + 2.66i)2-s + (−2.80 − 4.37i)3-s + 6.17i·4-s + (−9.55 + 5.80i)5-s + (4.17 − 19.1i)6-s + (9.35 − 9.35i)7-s + (4.84 − 4.84i)8-s + (−11.2 + 24.5i)9-s + (−40.8 − 10.0i)10-s + 34.1i·11-s + (27.0 − 17.3i)12-s + (2.82 + 2.82i)13-s + 49.8·14-s + (52.1 + 25.5i)15-s + 75.2·16-s + (−64.2 − 64.2i)17-s + ⋯
 L(s)  = 1 + (0.941 + 0.941i)2-s + (−0.539 − 0.841i)3-s + 0.772i·4-s + (−0.854 + 0.518i)5-s + (0.284 − 1.30i)6-s + (0.505 − 0.505i)7-s + (0.214 − 0.214i)8-s + (−0.417 + 0.908i)9-s + (−1.29 − 0.316i)10-s + 0.935i·11-s + (0.650 − 0.416i)12-s + (0.0601 + 0.0601i)13-s + 0.951·14-s + (0.898 + 0.439i)15-s + 1.17·16-s + (−0.916 − 0.916i)17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.533i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.846 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$15$$    =    $$3 \cdot 5$$ $$\varepsilon$$ = $0.846 - 0.533i$ motivic weight = $$3$$ character : $\chi_{15} (8, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 15,\ (\ :3/2),\ 0.846 - 0.533i)$ $L(2)$ $\approx$ $1.18094 + 0.340966i$ $L(\frac12)$ $\approx$ $1.18094 + 0.340966i$ $L(\frac{5}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{3,\;5\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{3,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 $$1 + (2.80 + 4.37i)T$$
5 $$1 + (9.55 - 5.80i)T$$
good2 $$1 + (-2.66 - 2.66i)T + 8iT^{2}$$
7 $$1 + (-9.35 + 9.35i)T - 343iT^{2}$$
11 $$1 - 34.1iT - 1.33e3T^{2}$$
13 $$1 + (-2.82 - 2.82i)T + 2.19e3iT^{2}$$
17 $$1 + (64.2 + 64.2i)T + 4.91e3iT^{2}$$
19 $$1 - 19.0iT - 6.85e3T^{2}$$
23 $$1 + (-51.4 + 51.4i)T - 1.21e4iT^{2}$$
29 $$1 + 50.5T + 2.43e4T^{2}$$
31 $$1 + 93.3T + 2.97e4T^{2}$$
37 $$1 + (161. - 161. i)T - 5.06e4iT^{2}$$
41 $$1 + 88.7iT - 6.89e4T^{2}$$
43 $$1 + (-176. - 176. i)T + 7.95e4iT^{2}$$
47 $$1 + (38.2 + 38.2i)T + 1.03e5iT^{2}$$
53 $$1 + (344. - 344. i)T - 1.48e5iT^{2}$$
59 $$1 - 421.T + 2.05e5T^{2}$$
61 $$1 - 2T + 2.26e5T^{2}$$
67 $$1 + (-430. + 430. i)T - 3.00e5iT^{2}$$
71 $$1 + 733. iT - 3.57e5T^{2}$$
73 $$1 + (348. + 348. i)T + 3.89e5iT^{2}$$
79 $$1 - 588. iT - 4.93e5T^{2}$$
83 $$1 + (-217. + 217. i)T - 5.71e5iT^{2}$$
89 $$1 - 1.27e3T + 7.04e5T^{2}$$
97 $$1 + (432. - 432. i)T - 9.12e5iT^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}