# Properties

 Label 2-56e2-1.1-c1-0-57 Degree $2$ Conductor $3136$ Sign $-1$ Analytic cond. $25.0410$ Root an. cond. $5.00410$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 3·9-s + 4·11-s − 8·23-s − 5·25-s − 2·29-s + 6·37-s − 12·43-s + 10·53-s + 4·67-s − 16·71-s − 8·79-s + 9·81-s − 12·99-s − 20·107-s − 18·109-s + 2·113-s + ⋯
 L(s)  = 1 − 9-s + 1.20·11-s − 1.66·23-s − 25-s − 0.371·29-s + 0.986·37-s − 1.82·43-s + 1.37·53-s + 0.488·67-s − 1.89·71-s − 0.900·79-s + 81-s − 1.20·99-s − 1.93·107-s − 1.72·109-s + 0.188·113-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3136$$    =    $$2^{6} \cdot 7^{2}$$ Sign: $-1$ Analytic conductor: $$25.0410$$ Root analytic conductor: $$5.00410$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{3136} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 3136,\ (\ :1/2),\ -1)$$

## 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$$F_p(T)$
bad2 $$1$$
7 $$1$$
good3 $$1 + p T^{2}$$
5 $$1 + p T^{2}$$
11 $$1 - 4 T + p T^{2}$$
13 $$1 + p T^{2}$$
17 $$1 + p T^{2}$$
19 $$1 + p T^{2}$$
23 $$1 + 8 T + p T^{2}$$
29 $$1 + 2 T + p T^{2}$$
31 $$1 + p T^{2}$$
37 $$1 - 6 T + p T^{2}$$
41 $$1 + p T^{2}$$
43 $$1 + 12 T + p T^{2}$$
47 $$1 + p T^{2}$$
53 $$1 - 10 T + p T^{2}$$
59 $$1 + p T^{2}$$
61 $$1 + p T^{2}$$
67 $$1 - 4 T + p T^{2}$$
71 $$1 + 16 T + p T^{2}$$
73 $$1 + p T^{2}$$
79 $$1 + 8 T + p T^{2}$$
83 $$1 + 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}$$