# Properties

 Label 2-15e2-1.1-c3-0-9 Degree $2$ Conductor $225$ Sign $-1$ Analytic cond. $13.2754$ Root an. cond. $3.64354$ Motivic weight $3$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 − 4·2-s + 8·4-s − 6·7-s − 32·11-s + 38·13-s + 24·14-s − 64·16-s + 26·17-s + 100·19-s + 128·22-s − 78·23-s − 152·26-s − 48·28-s + 50·29-s − 108·31-s + 256·32-s − 104·34-s − 266·37-s − 400·38-s − 22·41-s − 442·43-s − 256·44-s + 312·46-s − 514·47-s − 307·49-s + 304·52-s + 2·53-s + ⋯
 L(s)  = 1 − 1.41·2-s + 4-s − 0.323·7-s − 0.877·11-s + 0.810·13-s + 0.458·14-s − 16-s + 0.370·17-s + 1.20·19-s + 1.24·22-s − 0.707·23-s − 1.14·26-s − 0.323·28-s + 0.320·29-s − 0.625·31-s + 1.41·32-s − 0.524·34-s − 1.18·37-s − 1.70·38-s − 0.0838·41-s − 1.56·43-s − 0.877·44-s + 1.00·46-s − 1.59·47-s − 0.895·49-s + 0.810·52-s + 0.00518·53-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$225$$    =    $$3^{2} \cdot 5^{2}$$ Sign: $-1$ Analytic conductor: $$13.2754$$ Root analytic conductor: $$3.64354$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: $\chi_{225} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 225,\ (\ :3/2),\ -1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ 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^{2} T + p^{3} T^{2}$$
7 $$1 + 6 T + p^{3} T^{2}$$
11 $$1 + 32 T + p^{3} T^{2}$$
13 $$1 - 38 T + p^{3} T^{2}$$
17 $$1 - 26 T + p^{3} T^{2}$$
19 $$1 - 100 T + p^{3} T^{2}$$
23 $$1 + 78 T + p^{3} T^{2}$$
29 $$1 - 50 T + p^{3} T^{2}$$
31 $$1 + 108 T + p^{3} T^{2}$$
37 $$1 + 266 T + p^{3} T^{2}$$
41 $$1 + 22 T + p^{3} T^{2}$$
43 $$1 + 442 T + p^{3} T^{2}$$
47 $$1 + 514 T + p^{3} T^{2}$$
53 $$1 - 2 T + p^{3} T^{2}$$
59 $$1 + 500 T + p^{3} T^{2}$$
61 $$1 + 518 T + p^{3} T^{2}$$
67 $$1 + 126 T + p^{3} T^{2}$$
71 $$1 + 412 T + p^{3} T^{2}$$
73 $$1 - 878 T + p^{3} T^{2}$$
79 $$1 - 600 T + p^{3} T^{2}$$
83 $$1 - 282 T + p^{3} T^{2}$$
89 $$1 - 150 T + p^{3} T^{2}$$
97 $$1 + 386 T + p^{3} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$