# Properties

 Degree $2$ Conductor $25$ Sign $-0.894 - 0.447i$ Motivic weight $5$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 5.26i·2-s + 25.5i·3-s + 4.31·4-s − 134.·6-s − 131. i·7-s + 191. i·8-s − 408.·9-s + 290.·11-s + 110. i·12-s + 68.3i·13-s + 689.·14-s − 867.·16-s + 310. i·17-s − 2.14e3i·18-s + 2.13e3·19-s + ⋯
 L(s)  = 1 + 0.930i·2-s + 1.63i·3-s + 0.134·4-s − 1.52·6-s − 1.01i·7-s + 1.05i·8-s − 1.68·9-s + 0.722·11-s + 0.220i·12-s + 0.112i·13-s + 0.940·14-s − 0.847·16-s + 0.260i·17-s − 1.56i·18-s + 1.35·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$25$$    =    $$5^{2}$$ Sign: $-0.894 - 0.447i$ Motivic weight: $$5$$ Character: $\chi_{25} (24, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 25,\ (\ :5/2),\ -0.894 - 0.447i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.364966 + 1.54602i$$ $$L(\frac12)$$ $$\approx$$ $$0.364966 + 1.54602i$$ $$L(\frac{7}{2})$$ 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 - 5.26iT - 32T^{2}$$
3 $$1 - 25.5iT - 243T^{2}$$
7 $$1 + 131. iT - 1.68e4T^{2}$$
11 $$1 - 290.T + 1.61e5T^{2}$$
13 $$1 - 68.3iT - 3.71e5T^{2}$$
17 $$1 - 310. iT - 1.41e6T^{2}$$
19 $$1 - 2.13e3T + 2.47e6T^{2}$$
23 $$1 - 873. iT - 6.43e6T^{2}$$
29 $$1 - 2.58e3T + 2.05e7T^{2}$$
31 $$1 + 9.08e3T + 2.86e7T^{2}$$
37 $$1 + 3.99e3iT - 6.93e7T^{2}$$
41 $$1 - 1.69e4T + 1.15e8T^{2}$$
43 $$1 + 1.80e4iT - 1.47e8T^{2}$$
47 $$1 + 2.48e4iT - 2.29e8T^{2}$$
53 $$1 + 7.65e3iT - 4.18e8T^{2}$$
59 $$1 - 9.23e3T + 7.14e8T^{2}$$
61 $$1 - 3.32e3T + 8.44e8T^{2}$$
67 $$1 - 3.23e4iT - 1.35e9T^{2}$$
71 $$1 + 3.58e4T + 1.80e9T^{2}$$
73 $$1 + 2.65e4iT - 2.07e9T^{2}$$
79 $$1 + 7.17e4T + 3.07e9T^{2}$$
83 $$1 - 3.96e4iT - 3.93e9T^{2}$$
89 $$1 - 1.17e5T + 5.58e9T^{2}$$
97 $$1 + 2.18e4iT - 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$