# Properties

 Degree $2$ Conductor $2304$ Sign $0.577 - 0.816i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.41i·5-s + 2.82i·7-s + 4·11-s + 2·13-s − 1.41i·17-s − 5.65i·19-s + 4·23-s + 2.99·25-s + 7.07i·29-s − 8.48i·31-s − 4.00·35-s + 8·37-s − 4.24i·41-s + 11.3i·43-s − 12·47-s + ⋯
 L(s)  = 1 + 0.632i·5-s + 1.06i·7-s + 1.20·11-s + 0.554·13-s − 0.342i·17-s − 1.29i·19-s + 0.834·23-s + 0.599·25-s + 1.31i·29-s − 1.52i·31-s − 0.676·35-s + 1.31·37-s − 0.662i·41-s + 1.72i·43-s − 1.75·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2304$$    =    $$2^{8} \cdot 3^{2}$$ Sign: $0.577 - 0.816i$ Motivic weight: $$1$$ Character: $\chi_{2304} (2303, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2304,\ (\ :1/2),\ 0.577 - 0.816i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.038506609$$ $$L(\frac12)$$ $$\approx$$ $$2.038506609$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
good5 $$1 - 1.41iT - 5T^{2}$$
7 $$1 - 2.82iT - 7T^{2}$$
11 $$1 - 4T + 11T^{2}$$
13 $$1 - 2T + 13T^{2}$$
17 $$1 + 1.41iT - 17T^{2}$$
19 $$1 + 5.65iT - 19T^{2}$$
23 $$1 - 4T + 23T^{2}$$
29 $$1 - 7.07iT - 29T^{2}$$
31 $$1 + 8.48iT - 31T^{2}$$
37 $$1 - 8T + 37T^{2}$$
41 $$1 + 4.24iT - 41T^{2}$$
43 $$1 - 11.3iT - 43T^{2}$$
47 $$1 + 12T + 47T^{2}$$
53 $$1 - 12.7iT - 53T^{2}$$
59 $$1 + 59T^{2}$$
61 $$1 - 8T + 61T^{2}$$
67 $$1 + 5.65iT - 67T^{2}$$
71 $$1 - 4T + 71T^{2}$$
73 $$1 + 8T + 73T^{2}$$
79 $$1 + 2.82iT - 79T^{2}$$
83 $$1 + 12T + 83T^{2}$$
89 $$1 - 15.5iT - 89T^{2}$$
97 $$1 + 97T^{2}$$
show less
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−9.182796358858131062005746422852, −8.539105003596820625964827664326, −7.49112417903938353981118073297, −6.65999770087998053024576615956, −6.19019991054668863312000503695, −5.18977828004671580578011363272, −4.31004380098127609361863092683, −3.16424180665631368706856515975, −2.52494194985834679533180471482, −1.16084756680104319818946279339, 0.876164686237853542763896459170, 1.66296134316450211293625620685, 3.34591889857637317195337084234, 4.01354161046819685313686401912, 4.74871547394549170034787114716, 5.81178760462294483788582839207, 6.62178011092317198411646866205, 7.26508398200434080060145066892, 8.352174433618556756146477667705, 8.671132159022916897196033894620