# Properties

 Degree $2$ Conductor $25$ Sign $0.525 - 0.850i$ Motivic weight $8$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (11.8 + 11.8i)2-s + (11.8 − 11.8i)3-s + 26i·4-s + 282·6-s + (2.74e3 + 2.74e3i)7-s + (2.73e3 − 2.73e3i)8-s + 6.27e3i·9-s + 1.21e4·11-s + (308. + 308. i)12-s + (−2.42e3 + 2.42e3i)13-s + 6.51e4i·14-s + 7.15e4·16-s + (−7.69e4 − 7.69e4i)17-s + (−7.45e4 + 7.45e4i)18-s + 1.68e5i·19-s + ⋯
 L(s)  = 1 + (0.742 + 0.742i)2-s + (0.146 − 0.146i)3-s + 0.101i·4-s + 0.217·6-s + (1.14 + 1.14i)7-s + (0.666 − 0.666i)8-s + 0.957i·9-s + 0.828·11-s + (0.0148 + 0.0148i)12-s + (−0.0848 + 0.0848i)13-s + 1.69i·14-s + 1.09·16-s + (−0.921 − 0.921i)17-s + (−0.710 + 0.710i)18-s + 1.29i·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$25$$    =    $$5^{2}$$ Sign: $0.525 - 0.850i$ Motivic weight: $$8$$ Character: $\chi_{25} (7, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 25,\ (\ :4),\ 0.525 - 0.850i)$$

## Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$2.64012 + 1.47196i$$ $$L(\frac12)$$ $$\approx$$ $$2.64012 + 1.47196i$$ $$L(5)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1$$
good2 $$1 + (-11.8 - 11.8i)T + 256iT^{2}$$
3 $$1 + (-11.8 + 11.8i)T - 6.56e3iT^{2}$$
7 $$1 + (-2.74e3 - 2.74e3i)T + 5.76e6iT^{2}$$
11 $$1 - 1.21e4T + 2.14e8T^{2}$$
13 $$1 + (2.42e3 - 2.42e3i)T - 8.15e8iT^{2}$$
17 $$1 + (7.69e4 + 7.69e4i)T + 6.97e9iT^{2}$$
19 $$1 - 1.68e5iT - 1.69e10T^{2}$$
23 $$1 + (-2.21e5 + 2.21e5i)T - 7.83e10iT^{2}$$
29 $$1 + 6.66e5iT - 5.00e11T^{2}$$
31 $$1 + 1.04e6T + 8.52e11T^{2}$$
37 $$1 + (2.07e6 + 2.07e6i)T + 3.51e12iT^{2}$$
41 $$1 + 1.32e6T + 7.98e12T^{2}$$
43 $$1 + (-2.78e6 + 2.78e6i)T - 1.16e13iT^{2}$$
47 $$1 + (-3.63e6 - 3.63e6i)T + 2.38e13iT^{2}$$
53 $$1 + (3.14e6 - 3.14e6i)T - 6.22e13iT^{2}$$
59 $$1 - 6.49e6iT - 1.46e14T^{2}$$
61 $$1 + 1.43e7T + 1.91e14T^{2}$$
67 $$1 + (1.14e7 + 1.14e7i)T + 4.06e14iT^{2}$$
71 $$1 + 2.30e7T + 6.45e14T^{2}$$
73 $$1 + (-1.74e7 + 1.74e7i)T - 8.06e14iT^{2}$$
79 $$1 + 2.76e6iT - 1.51e15T^{2}$$
83 $$1 + (1.15e7 - 1.15e7i)T - 2.25e15iT^{2}$$
89 $$1 - 2.61e7iT - 3.93e15T^{2}$$
97 $$1 + (8.09e7 + 8.09e7i)T + 7.83e15iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$