# Properties

 Degree $2$ Conductor $25$ Sign $-1$ Motivic weight $9$ Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 31.7·2-s − 268.·3-s + 498.·4-s + 8.53e3·6-s + 637.·7-s + 440.·8-s + 5.24e4·9-s − 4.90e4·11-s − 1.33e5·12-s + 7.27e4·13-s − 2.02e4·14-s − 2.69e5·16-s + 6.73e4·17-s − 1.66e6·18-s + 3.41e5·19-s − 1.71e5·21-s + 1.55e6·22-s − 1.34e5·23-s − 1.18e5·24-s − 2.31e6·26-s − 8.81e6·27-s + 3.17e5·28-s + 4.45e6·29-s + 4.56e5·31-s + 8.32e6·32-s + 1.31e7·33-s − 2.13e6·34-s + ⋯
 L(s)  = 1 − 1.40·2-s − 1.91·3-s + 0.972·4-s + 2.68·6-s + 0.100·7-s + 0.0380·8-s + 2.66·9-s − 1.00·11-s − 1.86·12-s + 0.706·13-s − 0.140·14-s − 1.02·16-s + 0.195·17-s − 3.74·18-s + 0.600·19-s − 0.192·21-s + 1.41·22-s − 0.100·23-s − 0.0727·24-s − 0.991·26-s − 3.19·27-s + 0.0975·28-s + 1.17·29-s + 0.0887·31-s + 1.40·32-s + 1.93·33-s − 0.274·34-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$25$$    =    $$5^{2}$$ Sign: $-1$ Motivic weight: $$9$$ Character: $\chi_{25} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 25,\ (\ :9/2),\ -1)$$

## Particular Values

 $$L(5)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{11}{2})$$ 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 + 31.7T + 512T^{2}$$
3 $$1 + 268.T + 1.96e4T^{2}$$
7 $$1 - 637.T + 4.03e7T^{2}$$
11 $$1 + 4.90e4T + 2.35e9T^{2}$$
13 $$1 - 7.27e4T + 1.06e10T^{2}$$
17 $$1 - 6.73e4T + 1.18e11T^{2}$$
19 $$1 - 3.41e5T + 3.22e11T^{2}$$
23 $$1 + 1.34e5T + 1.80e12T^{2}$$
29 $$1 - 4.45e6T + 1.45e13T^{2}$$
31 $$1 - 4.56e5T + 2.64e13T^{2}$$
37 $$1 - 1.30e7T + 1.29e14T^{2}$$
41 $$1 + 2.56e7T + 3.27e14T^{2}$$
43 $$1 - 3.42e6T + 5.02e14T^{2}$$
47 $$1 + 3.39e7T + 1.11e15T^{2}$$
53 $$1 + 8.42e7T + 3.29e15T^{2}$$
59 $$1 + 7.46e7T + 8.66e15T^{2}$$
61 $$1 - 1.78e8T + 1.16e16T^{2}$$
67 $$1 - 6.94e7T + 2.72e16T^{2}$$
71 $$1 + 2.07e8T + 4.58e16T^{2}$$
73 $$1 + 3.02e8T + 5.88e16T^{2}$$
79 $$1 - 3.72e8T + 1.19e17T^{2}$$
83 $$1 - 4.50e8T + 1.86e17T^{2}$$
89 $$1 - 5.82e7T + 3.50e17T^{2}$$
97 $$1 - 7.85e8T + 7.60e17T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$