# Properties

 Degree $2$ Conductor $1344$ Sign $0.406 - 0.913i$ Motivic weight $3$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 3i·3-s + 12.5·5-s + (18.2 + 2.88i)7-s − 9·9-s + 55.4·11-s + 92.1·13-s + 37.5i·15-s + 118. i·17-s + 155. i·19-s + (−8.66 + 54.8i)21-s − 125. i·23-s + 31.6·25-s − 27i·27-s + 131. i·29-s − 66.0·31-s + ⋯
 L(s)  = 1 + 0.577i·3-s + 1.11·5-s + (0.987 + 0.155i)7-s − 0.333·9-s + 1.51·11-s + 1.96·13-s + 0.646i·15-s + 1.68i·17-s + 1.87i·19-s + (−0.0900 + 0.570i)21-s − 1.13i·23-s + 0.253·25-s − 0.192i·27-s + 0.842i·29-s − 0.382·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.406 - 0.913i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.406 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1344$$    =    $$2^{6} \cdot 3 \cdot 7$$ Sign: $0.406 - 0.913i$ Motivic weight: $$3$$ Character: $\chi_{1344} (223, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1344,\ (\ :3/2),\ 0.406 - 0.913i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$3.881688195$$ $$L(\frac12)$$ $$\approx$$ $$3.881688195$$ $$L(\frac{5}{2})$$ 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 - 3iT$$
7 $$1 + (-18.2 - 2.88i)T$$
good5 $$1 - 12.5T + 125T^{2}$$
11 $$1 - 55.4T + 1.33e3T^{2}$$
13 $$1 - 92.1T + 2.19e3T^{2}$$
17 $$1 - 118. iT - 4.91e3T^{2}$$
19 $$1 - 155. iT - 6.85e3T^{2}$$
23 $$1 + 125. iT - 1.21e4T^{2}$$
29 $$1 - 131. iT - 2.43e4T^{2}$$
31 $$1 + 66.0T + 2.97e4T^{2}$$
37 $$1 + 147. iT - 5.06e4T^{2}$$
41 $$1 - 20.3iT - 6.89e4T^{2}$$
43 $$1 + 355.T + 7.95e4T^{2}$$
47 $$1 + 79.5T + 1.03e5T^{2}$$
53 $$1 + 463. iT - 1.48e5T^{2}$$
59 $$1 + 580. iT - 2.05e5T^{2}$$
61 $$1 - 587.T + 2.26e5T^{2}$$
67 $$1 + 496.T + 3.00e5T^{2}$$
71 $$1 - 232. iT - 3.57e5T^{2}$$
73 $$1 + 551. iT - 3.89e5T^{2}$$
79 $$1 + 437. iT - 4.93e5T^{2}$$
83 $$1 - 191. iT - 5.71e5T^{2}$$
89 $$1 - 93.7iT - 7.04e5T^{2}$$
97 $$1 + 758. iT - 9.12e5T^{2}$$
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$$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

−9.350759261943019966450088895307, −8.482938885727058851038782358576, −8.279516896585644390813965598056, −6.52109720132618743130889278174, −6.09140567393966279587335938932, −5.36263569958889021368168257620, −4.03602542661140418654681815044, −3.63535765947389657674283400020, −1.71014046658485481046886993298, −1.50100015198367664218127040220, 0.993865546873576556899756578305, 1.51075988532183141874963265052, 2.67522256887267913096609204199, 3.92245205113131901937446173575, 5.01904455863587628658554802998, 5.85939757803443469670503887843, 6.64229657008665734265944656749, 7.28766923212592547665006921634, 8.448642194719918332120695996889, 9.091527620121302848341469018100