# Properties

 Label 2-95e2-1.1-c1-0-237 Degree $2$ Conductor $9025$ Sign $-1$ Analytic cond. $72.0649$ Root an. cond. $8.48910$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 − 2·4-s − 3·7-s − 3·9-s − 5·11-s + 4·16-s + 7·17-s + 4·23-s + 6·28-s + 6·36-s + 43-s + 10·44-s − 13·47-s + 2·49-s + 15·61-s + 9·63-s − 8·64-s − 14·68-s + 11·73-s + 15·77-s + 9·81-s + 16·83-s − 8·92-s + 15·99-s + 10·101-s − 12·112-s − 21·119-s + ⋯
 L(s)  = 1 − 4-s − 1.13·7-s − 9-s − 1.50·11-s + 16-s + 1.69·17-s + 0.834·23-s + 1.13·28-s + 36-s + 0.152·43-s + 1.50·44-s − 1.89·47-s + 2/7·49-s + 1.92·61-s + 1.13·63-s − 64-s − 1.69·68-s + 1.28·73-s + 1.70·77-s + 81-s + 1.75·83-s − 0.834·92-s + 1.50·99-s + 0.995·101-s − 1.13·112-s − 1.92·119-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$9025$$    =    $$5^{2} \cdot 19^{2}$$ Sign: $-1$ Analytic conductor: $$72.0649$$ Root analytic conductor: $$8.48910$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 9025,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1$$
19 $$1$$
good2 $$1 + p T^{2}$$
3 $$1 + p T^{2}$$
7 $$1 + 3 T + p T^{2}$$
11 $$1 + 5 T + p T^{2}$$
13 $$1 + p T^{2}$$
17 $$1 - 7 T + p T^{2}$$
23 $$1 - 4 T + p T^{2}$$
29 $$1 + p T^{2}$$
31 $$1 + p T^{2}$$
37 $$1 + p T^{2}$$
41 $$1 + p T^{2}$$
43 $$1 - T + p T^{2}$$
47 $$1 + 13 T + p T^{2}$$
53 $$1 + p T^{2}$$
59 $$1 + p T^{2}$$
61 $$1 - 15 T + p T^{2}$$
67 $$1 + p T^{2}$$
71 $$1 + p T^{2}$$
73 $$1 - 11 T + p T^{2}$$
79 $$1 + p T^{2}$$
83 $$1 - 16 T + p T^{2}$$
89 $$1 + p T^{2}$$
97 $$1 + p T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$