# Properties

 Degree $2$ Conductor $25$ Sign $0.447 + 0.894i$ Motivic weight $33$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.78e5i·2-s + 8.55e7i·3-s − 2.34e10·4-s + 1.52e13·6-s + 8.09e13i·7-s + 2.64e15i·8-s − 1.75e15·9-s + 1.95e17·11-s − 2.00e18i·12-s − 2.59e18i·13-s + 1.44e19·14-s + 2.72e20·16-s + 1.52e20i·17-s + 3.13e20i·18-s − 1.39e21·19-s + ⋯
 L(s)  = 1 − 1.92i·2-s + 1.14i·3-s − 2.72·4-s + 2.21·6-s + 0.920i·7-s + 3.32i·8-s − 0.315·9-s + 1.28·11-s − 3.12i·12-s − 1.08i·13-s + 1.77·14-s + 3.69·16-s + 0.758i·17-s + 0.608i·18-s − 1.10·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(34-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$25$$    =    $$5^{2}$$ Sign: $0.447 + 0.894i$ Motivic weight: $$33$$ Character: $\chi_{25} (24, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 25,\ (\ :33/2),\ 0.447 + 0.894i)$$

## Particular Values

 $$L(17)$$ $$\approx$$ $$1.820403983$$ $$L(\frac12)$$ $$\approx$$ $$1.820403983$$ $$L(\frac{35}{2})$$ 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 + 1.78e5iT - 8.58e9T^{2}$$
3 $$1 - 8.55e7iT - 5.55e15T^{2}$$
7 $$1 - 8.09e13iT - 7.73e27T^{2}$$
11 $$1 - 1.95e17T + 2.32e34T^{2}$$
13 $$1 + 2.59e18iT - 5.75e36T^{2}$$
17 $$1 - 1.52e20iT - 4.02e40T^{2}$$
19 $$1 + 1.39e21T + 1.58e42T^{2}$$
23 $$1 + 3.40e22iT - 8.65e44T^{2}$$
29 $$1 - 1.49e24T + 1.81e48T^{2}$$
31 $$1 + 1.38e24T + 1.64e49T^{2}$$
37 $$1 + 1.15e26iT - 5.63e51T^{2}$$
41 $$1 + 6.35e25T + 1.66e53T^{2}$$
43 $$1 + 2.64e26iT - 8.02e53T^{2}$$
47 $$1 + 1.03e27iT - 1.51e55T^{2}$$
53 $$1 - 5.20e28iT - 7.96e56T^{2}$$
59 $$1 - 6.17e28T + 2.74e58T^{2}$$
61 $$1 + 1.21e29T + 8.23e58T^{2}$$
67 $$1 - 2.87e29iT - 1.82e60T^{2}$$
71 $$1 + 2.84e30T + 1.23e61T^{2}$$
73 $$1 + 2.67e30iT - 3.08e61T^{2}$$
79 $$1 - 7.84e30T + 4.18e62T^{2}$$
83 $$1 - 1.28e31iT - 2.13e63T^{2}$$
89 $$1 - 1.56e32T + 2.13e64T^{2}$$
97 $$1 + 4.84e32iT - 3.65e65T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$