# Properties

 Label 2-304-76.75-c1-0-8 Degree $2$ Conductor $304$ Sign $i$ Analytic cond. $2.42745$ Root an. cond. $1.55802$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 5-s − 4.35i·7-s − 3·9-s − 4.35i·11-s + 7·17-s + 4.35i·19-s − 8.71i·23-s − 4·25-s + 4.35i·35-s + 13.0i·43-s + 3·45-s − 4.35i·47-s − 12.0·49-s + 4.35i·55-s + 15·61-s + ⋯
 L(s)  = 1 − 0.447·5-s − 1.64i·7-s − 9-s − 1.31i·11-s + 1.69·17-s + 0.999i·19-s − 1.81i·23-s − 0.800·25-s + 0.736i·35-s + 1.99i·43-s + 0.447·45-s − 0.635i·47-s − 1.71·49-s + 0.587i·55-s + 1.92·61-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$304$$    =    $$2^{4} \cdot 19$$ Sign: $i$ Analytic conductor: $$2.42745$$ Root analytic conductor: $$1.55802$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{304} (303, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 304,\ (\ :1/2),\ i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.709639 - 0.709639i$$ $$L(\frac12)$$ $$\approx$$ $$0.709639 - 0.709639i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
19 $$1 - 4.35iT$$
good3 $$1 + 3T^{2}$$
5 $$1 + T + 5T^{2}$$
7 $$1 + 4.35iT - 7T^{2}$$
11 $$1 + 4.35iT - 11T^{2}$$
13 $$1 - 13T^{2}$$
17 $$1 - 7T + 17T^{2}$$
23 $$1 + 8.71iT - 23T^{2}$$
29 $$1 - 29T^{2}$$
31 $$1 + 31T^{2}$$
37 $$1 - 37T^{2}$$
41 $$1 - 41T^{2}$$
43 $$1 - 13.0iT - 43T^{2}$$
47 $$1 + 4.35iT - 47T^{2}$$
53 $$1 - 53T^{2}$$
59 $$1 + 59T^{2}$$
61 $$1 - 15T + 61T^{2}$$
67 $$1 + 67T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 - 11T + 73T^{2}$$
79 $$1 + 79T^{2}$$
83 $$1 - 8.71iT - 83T^{2}$$
89 $$1 - 89T^{2}$$
97 $$1 - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$