# Properties

 Label 2-48e2-1.1-c3-0-70 Degree $2$ Conductor $2304$ Sign $-1$ Analytic cond. $135.940$ Root an. cond. $11.6593$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 10.3·5-s − 3.46·7-s − 55.4·13-s + 90·17-s − 116·19-s + 103.·23-s − 17·25-s + 259.·29-s + 301.·31-s + 36·35-s + 34.6·37-s − 54·41-s − 20·43-s + 394.·47-s − 331·49-s − 488.·53-s + 324·59-s + 575.·61-s + 576·65-s + 116·67-s − 1.10e3·71-s − 1.10e3·73-s + 148.·79-s − 1.15e3·83-s − 935.·85-s + 918·89-s + 191.·91-s + ⋯
 L(s)  = 1 − 0.929·5-s − 0.187·7-s − 1.18·13-s + 1.28·17-s − 1.40·19-s + 0.942·23-s − 0.136·25-s + 1.66·29-s + 1.74·31-s + 0.173·35-s + 0.153·37-s − 0.205·41-s − 0.0709·43-s + 1.22·47-s − 0.965·49-s − 1.26·53-s + 0.714·59-s + 1.20·61-s + 1.09·65-s + 0.211·67-s − 1.84·71-s − 1.77·73-s + 0.212·79-s − 1.52·83-s − 1.19·85-s + 1.09·89-s + 0.221·91-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2304$$    =    $$2^{8} \cdot 3^{2}$$ Sign: $-1$ Analytic conductor: $$135.940$$ Root analytic conductor: $$11.6593$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 2304,\ (\ :3/2),\ -1)$$

## 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$$F_p(T)$
bad2 $$1$$
3 $$1$$
good5 $$1 + 10.3T + 125T^{2}$$
7 $$1 + 3.46T + 343T^{2}$$
11 $$1 + 1.33e3T^{2}$$
13 $$1 + 55.4T + 2.19e3T^{2}$$
17 $$1 - 90T + 4.91e3T^{2}$$
19 $$1 + 116T + 6.85e3T^{2}$$
23 $$1 - 103.T + 1.21e4T^{2}$$
29 $$1 - 259.T + 2.43e4T^{2}$$
31 $$1 - 301.T + 2.97e4T^{2}$$
37 $$1 - 34.6T + 5.06e4T^{2}$$
41 $$1 + 54T + 6.89e4T^{2}$$
43 $$1 + 20T + 7.95e4T^{2}$$
47 $$1 - 394.T + 1.03e5T^{2}$$
53 $$1 + 488.T + 1.48e5T^{2}$$
59 $$1 - 324T + 2.05e5T^{2}$$
61 $$1 - 575.T + 2.26e5T^{2}$$
67 $$1 - 116T + 3.00e5T^{2}$$
71 $$1 + 1.10e3T + 3.57e5T^{2}$$
73 $$1 + 1.10e3T + 3.89e5T^{2}$$
79 $$1 - 148.T + 4.93e5T^{2}$$
83 $$1 + 1.15e3T + 5.71e5T^{2}$$
89 $$1 - 918T + 7.04e5T^{2}$$
97 $$1 - 190T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$