# Properties

 Degree 2 Conductor $2 \cdot 5$ Sign $0.973 + 0.229i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1 − i)2-s + (−2 + 2i)3-s + 2i·4-s − 5i·5-s + 4·6-s + (2 + 2i)7-s + (2 − 2i)8-s + i·9-s + (−5 + 5i)10-s − 8·11-s + (−4 − 4i)12-s + (3 − 3i)13-s − 4i·14-s + (10 + 10i)15-s − 4·16-s + (7 + 7i)17-s + ⋯
 L(s)  = 1 + (−0.5 − 0.5i)2-s + (−0.666 + 0.666i)3-s + 0.5i·4-s − i·5-s + 0.666·6-s + (0.285 + 0.285i)7-s + (0.250 − 0.250i)8-s + 0.111i·9-s + (−0.5 + 0.5i)10-s − 0.727·11-s + (−0.333 − 0.333i)12-s + (0.230 − 0.230i)13-s − 0.285i·14-s + (0.666 + 0.666i)15-s − 0.250·16-s + (0.411 + 0.411i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$10$$    =    $$2 \cdot 5$$ $$\varepsilon$$ = $0.973 + 0.229i$ motivic weight = $$2$$ character : $\chi_{10} (7, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 10,\ (\ :1),\ 0.973 + 0.229i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$0.485972 - 0.0565836i$$ $$L(\frac12)$$ $$\approx$$ $$0.485972 - 0.0565836i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;5\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + (1 + i)T$$
5 $$1 + 5iT$$
good3 $$1 + (2 - 2i)T - 9iT^{2}$$
7 $$1 + (-2 - 2i)T + 49iT^{2}$$
11 $$1 + 8T + 121T^{2}$$
13 $$1 + (-3 + 3i)T - 169iT^{2}$$
17 $$1 + (-7 - 7i)T + 289iT^{2}$$
19 $$1 + 20iT - 361T^{2}$$
23 $$1 + (2 - 2i)T - 529iT^{2}$$
29 $$1 - 40iT - 841T^{2}$$
31 $$1 - 52T + 961T^{2}$$
37 $$1 + (3 + 3i)T + 1.36e3iT^{2}$$
41 $$1 + 8T + 1.68e3T^{2}$$
43 $$1 + (42 - 42i)T - 1.84e3iT^{2}$$
47 $$1 + (18 + 18i)T + 2.20e3iT^{2}$$
53 $$1 + (-53 + 53i)T - 2.80e3iT^{2}$$
59 $$1 + 20iT - 3.48e3T^{2}$$
61 $$1 + 48T + 3.72e3T^{2}$$
67 $$1 + (-62 - 62i)T + 4.48e3iT^{2}$$
71 $$1 + 28T + 5.04e3T^{2}$$
73 $$1 + (47 - 47i)T - 5.32e3iT^{2}$$
79 $$1 - 6.24e3T^{2}$$
83 $$1 + (-18 + 18i)T - 6.88e3iT^{2}$$
89 $$1 - 80iT - 7.92e3T^{2}$$
97 $$1 + (63 + 63i)T + 9.40e3iT^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}