# Properties

 Label 2-304-19.18-c4-0-7 Degree $2$ Conductor $304$ Sign $1$ Analytic cond. $31.4244$ Root an. cond. $5.60575$ Motivic weight $4$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 49.4·5-s − 93.1·7-s + 81·9-s − 173.·11-s − 219.·17-s − 361·19-s + 158·23-s + 1.82e3·25-s + 4.60e3·35-s + 800.·43-s − 4.00e3·45-s − 3.07e3·47-s + 6.27e3·49-s + 8.56e3·55-s − 7.41e3·61-s − 7.54e3·63-s + 1.90e3·73-s + 1.61e4·77-s + 6.56e3·81-s + 5.67e3·83-s + 1.08e4·85-s + 1.78e4·95-s − 1.40e4·99-s − 9.99e3·101-s − 7.81e3·115-s + 2.04e4·119-s + ⋯
 L(s)  = 1 − 1.97·5-s − 1.90·7-s + 9-s − 1.43·11-s − 0.760·17-s − 19-s + 0.298·23-s + 2.91·25-s + 3.76·35-s + 0.433·43-s − 1.97·45-s − 1.39·47-s + 2.61·49-s + 2.83·55-s − 1.99·61-s − 1.90·63-s + 0.356·73-s + 2.71·77-s + 81-s + 0.824·83-s + 1.50·85-s + 1.97·95-s − 1.43·99-s − 0.980·101-s − 0.591·115-s + 1.44·119-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$304$$    =    $$2^{4} \cdot 19$$ Sign: $1$ Analytic conductor: $$31.4244$$ Root analytic conductor: $$5.60575$$ Motivic weight: $$4$$ Rational: no Arithmetic: yes Character: $\chi_{304} (113, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 304,\ (\ :2),\ 1)$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$0.3189813896$$ $$L(\frac12)$$ $$\approx$$ $$0.3189813896$$ $$L(3)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
19 $$1 + 361T$$
good3 $$1 - 81T^{2}$$
5 $$1 + 49.4T + 625T^{2}$$
7 $$1 + 93.1T + 2.40e3T^{2}$$
11 $$1 + 173.T + 1.46e4T^{2}$$
13 $$1 - 2.85e4T^{2}$$
17 $$1 + 219.T + 8.35e4T^{2}$$
23 $$1 - 158T + 2.79e5T^{2}$$
29 $$1 - 7.07e5T^{2}$$
31 $$1 - 9.23e5T^{2}$$
37 $$1 - 1.87e6T^{2}$$
41 $$1 - 2.82e6T^{2}$$
43 $$1 - 800.T + 3.41e6T^{2}$$
47 $$1 + 3.07e3T + 4.87e6T^{2}$$
53 $$1 - 7.89e6T^{2}$$
59 $$1 - 1.21e7T^{2}$$
61 $$1 + 7.41e3T + 1.38e7T^{2}$$
67 $$1 - 2.01e7T^{2}$$
71 $$1 - 2.54e7T^{2}$$
73 $$1 - 1.90e3T + 2.83e7T^{2}$$
79 $$1 - 3.89e7T^{2}$$
83 $$1 - 5.67e3T + 4.74e7T^{2}$$
89 $$1 - 6.27e7T^{2}$$
97 $$1 - 8.85e7T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$