# Properties

 Label 2-304-1.1-c3-0-22 Degree $2$ Conductor $304$ Sign $-1$ Analytic cond. $17.9365$ Root an. cond. $4.23516$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.33·3-s + 18.4·5-s − 10.3·7-s − 25.2·9-s − 50.4·11-s − 61.8·13-s − 24.5·15-s + 68.1·17-s + 19·19-s + 13.8·21-s − 145.·23-s + 215.·25-s + 69.5·27-s + 42.6·29-s − 91.6·31-s + 67.1·33-s − 191.·35-s − 400.·37-s + 82.3·39-s − 123.·41-s − 449.·43-s − 465.·45-s + 453.·47-s − 235.·49-s − 90.7·51-s + 437.·53-s − 930.·55-s + ⋯
 L(s)  = 1 − 0.256·3-s + 1.64·5-s − 0.560·7-s − 0.934·9-s − 1.38·11-s − 1.31·13-s − 0.422·15-s + 0.971·17-s + 0.229·19-s + 0.143·21-s − 1.32·23-s + 1.72·25-s + 0.495·27-s + 0.272·29-s − 0.531·31-s + 0.354·33-s − 0.925·35-s − 1.78·37-s + 0.338·39-s − 0.469·41-s − 1.59·43-s − 1.54·45-s + 1.40·47-s − 0.685·49-s − 0.249·51-s + 1.13·53-s − 2.28·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$304$$    =    $$2^{4} \cdot 19$$ Sign: $-1$ Analytic conductor: $$17.9365$$ Root analytic conductor: $$4.23516$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 304,\ (\ :3/2),\ -1)$$

## 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$$
19 $$1 - 19T$$
good3 $$1 + 1.33T + 27T^{2}$$
5 $$1 - 18.4T + 125T^{2}$$
7 $$1 + 10.3T + 343T^{2}$$
11 $$1 + 50.4T + 1.33e3T^{2}$$
13 $$1 + 61.8T + 2.19e3T^{2}$$
17 $$1 - 68.1T + 4.91e3T^{2}$$
23 $$1 + 145.T + 1.21e4T^{2}$$
29 $$1 - 42.6T + 2.43e4T^{2}$$
31 $$1 + 91.6T + 2.97e4T^{2}$$
37 $$1 + 400.T + 5.06e4T^{2}$$
41 $$1 + 123.T + 6.89e4T^{2}$$
43 $$1 + 449.T + 7.95e4T^{2}$$
47 $$1 - 453.T + 1.03e5T^{2}$$
53 $$1 - 437.T + 1.48e5T^{2}$$
59 $$1 - 159.T + 2.05e5T^{2}$$
61 $$1 + 476.T + 2.26e5T^{2}$$
67 $$1 - 629.T + 3.00e5T^{2}$$
71 $$1 + 471.T + 3.57e5T^{2}$$
73 $$1 + 725.T + 3.89e5T^{2}$$
79 $$1 - 1.05e3T + 4.93e5T^{2}$$
83 $$1 - 726.T + 5.71e5T^{2}$$
89 $$1 + 468.T + 7.04e5T^{2}$$
97 $$1 + 891.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$