# Properties

 Degree 2 Conductor $5^{2}$ Sign $0.793 + 0.608i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.22 − 1.22i)2-s + (−1.22 − 1.22i)3-s + 1.00i·4-s − 2.99·6-s + (−4.89 + 4.89i)7-s + (6.12 + 6.12i)8-s − 6i·9-s − 3·11-s + (1.22 − 1.22i)12-s + (−7.34 − 7.34i)13-s + 11.9i·14-s + 10.9·16-s + (13.4 − 13.4i)17-s + (−7.34 − 7.34i)18-s − 5i·19-s + ⋯
 L(s)  = 1 + (0.612 − 0.612i)2-s + (−0.408 − 0.408i)3-s + 0.250i·4-s − 0.499·6-s + (−0.699 + 0.699i)7-s + (0.765 + 0.765i)8-s − 0.666i·9-s − 0.272·11-s + (0.102 − 0.102i)12-s + (−0.565 − 0.565i)13-s + 0.857i·14-s + 0.687·16-s + (0.792 − 0.792i)17-s + (−0.408 − 0.408i)18-s − 0.263i·19-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$25$$    =    $$5^{2}$$ $$\varepsilon$$ = $0.793 + 0.608i$ motivic weight = $$2$$ character : $\chi_{25} (18, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 25,\ (\ :1),\ 0.793 + 0.608i)$ $L(\frac{3}{2})$ $\approx$ $1.00672 - 0.341600i$ $L(\frac12)$ $\approx$ $1.00672 - 0.341600i$ $L(2)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 5$, $$F_p$$ is a polynomial of degree 2. If $p = 5$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 $$1$$
good2 $$1 + (-1.22 + 1.22i)T - 4iT^{2}$$
3 $$1 + (1.22 + 1.22i)T + 9iT^{2}$$
7 $$1 + (4.89 - 4.89i)T - 49iT^{2}$$
11 $$1 + 3T + 121T^{2}$$
13 $$1 + (7.34 + 7.34i)T + 169iT^{2}$$
17 $$1 + (-13.4 + 13.4i)T - 289iT^{2}$$
19 $$1 + 5iT - 361T^{2}$$
23 $$1 + (-17.1 - 17.1i)T + 529iT^{2}$$
29 $$1 - 30iT - 841T^{2}$$
31 $$1 + 38T + 961T^{2}$$
37 $$1 + (-19.5 + 19.5i)T - 1.36e3iT^{2}$$
41 $$1 - 57T + 1.68e3T^{2}$$
43 $$1 + (-4.89 - 4.89i)T + 1.84e3iT^{2}$$
47 $$1 + (-7.34 + 7.34i)T - 2.20e3iT^{2}$$
53 $$1 + (31.8 + 31.8i)T + 2.80e3iT^{2}$$
59 $$1 + 90iT - 3.48e3T^{2}$$
61 $$1 + 28T + 3.72e3T^{2}$$
67 $$1 + (47.7 - 47.7i)T - 4.48e3iT^{2}$$
71 $$1 - 42T + 5.04e3T^{2}$$
73 $$1 + (13.4 + 13.4i)T + 5.32e3iT^{2}$$
79 $$1 - 80iT - 6.24e3T^{2}$$
83 $$1 + (111. + 111. i)T + 6.88e3iT^{2}$$
89 $$1 - 15iT - 7.92e3T^{2}$$
97 $$1 + (53.8 - 53.8i)T - 9.40e3iT^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}