# Properties

 Label 2-45e2-1.1-c1-0-68 Degree $2$ Conductor $2025$ Sign $-1$ Analytic cond. $16.1697$ Root an. cond. $4.02115$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 + 2·2-s + 2·4-s − 5·11-s − 4·13-s − 4·16-s − 4·17-s − 5·19-s − 10·22-s + 6·23-s − 8·26-s + 5·29-s − 9·31-s − 8·32-s − 8·34-s + 10·37-s − 10·38-s − 7·41-s + 2·43-s − 10·44-s + 12·46-s + 2·47-s − 7·49-s − 8·52-s + 8·53-s + 10·58-s + 59-s − 2·61-s + ⋯
 L(s)  = 1 + 1.41·2-s + 4-s − 1.50·11-s − 1.10·13-s − 16-s − 0.970·17-s − 1.14·19-s − 2.13·22-s + 1.25·23-s − 1.56·26-s + 0.928·29-s − 1.61·31-s − 1.41·32-s − 1.37·34-s + 1.64·37-s − 1.62·38-s − 1.09·41-s + 0.304·43-s − 1.50·44-s + 1.76·46-s + 0.291·47-s − 49-s − 1.10·52-s + 1.09·53-s + 1.31·58-s + 0.130·59-s − 0.256·61-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2025$$    =    $$3^{4} \cdot 5^{2}$$ Sign: $-1$ Analytic conductor: $$16.1697$$ Root analytic conductor: $$4.02115$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 2025,\ (\ :1/2),\ -1)$$

## 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$$F_p(T)$
bad3 $$1$$
5 $$1$$
good2 $$1 - p T + p T^{2}$$
7 $$1 + p T^{2}$$
11 $$1 + 5 T + p T^{2}$$
13 $$1 + 4 T + p T^{2}$$
17 $$1 + 4 T + p T^{2}$$
19 $$1 + 5 T + p T^{2}$$
23 $$1 - 6 T + p T^{2}$$
29 $$1 - 5 T + p T^{2}$$
31 $$1 + 9 T + p T^{2}$$
37 $$1 - 10 T + p T^{2}$$
41 $$1 + 7 T + p T^{2}$$
43 $$1 - 2 T + p T^{2}$$
47 $$1 - 2 T + p T^{2}$$
53 $$1 - 8 T + p T^{2}$$
59 $$1 - T + p T^{2}$$
61 $$1 + 2 T + p T^{2}$$
67 $$1 + 6 T + p T^{2}$$
71 $$1 + T + p T^{2}$$
73 $$1 - 8 T + p T^{2}$$
79 $$1 - 12 T + p T^{2}$$
83 $$1 - 6 T + p T^{2}$$
89 $$1 - 9 T + p T^{2}$$
97 $$1 + 14 T + p T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$