# Properties

 Label 2-1344-56.27-c3-0-18 Degree $2$ Conductor $1344$ Sign $-0.650 - 0.759i$ Analytic cond. $79.2985$ Root an. cond. $8.90497$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 3i·3-s + 1.64·5-s + (−15.2 − 10.4i)7-s − 9·9-s + 20.3·11-s + 13.0·13-s + 4.92i·15-s − 23.9i·17-s + 87.7i·19-s + (31.4 − 45.8i)21-s − 73.6i·23-s − 122.·25-s − 27i·27-s + 58.9i·29-s + 124.·31-s + ⋯
 L(s)  = 1 + 0.577i·3-s + 0.146·5-s + (−0.824 − 0.565i)7-s − 0.333·9-s + 0.557·11-s + 0.279·13-s + 0.0847i·15-s − 0.341i·17-s + 1.05i·19-s + (0.326 − 0.476i)21-s − 0.667i·23-s − 0.978·25-s − 0.192i·27-s + 0.377i·29-s + 0.723·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.650 - 0.759i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.650 - 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1344$$    =    $$2^{6} \cdot 3 \cdot 7$$ Sign: $-0.650 - 0.759i$ Analytic conductor: $$79.2985$$ Root analytic conductor: $$8.90497$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{1344} (223, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1344,\ (\ :3/2),\ -0.650 - 0.759i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.025166091$$ $$L(\frac12)$$ $$\approx$$ $$1.025166091$$ $$L(\frac{5}{2})$$ 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 + (15.2 + 10.4i)T$$
good5 $$1 - 1.64T + 125T^{2}$$
11 $$1 - 20.3T + 1.33e3T^{2}$$
13 $$1 - 13.0T + 2.19e3T^{2}$$
17 $$1 + 23.9iT - 4.91e3T^{2}$$
19 $$1 - 87.7iT - 6.85e3T^{2}$$
23 $$1 + 73.6iT - 1.21e4T^{2}$$
29 $$1 - 58.9iT - 2.43e4T^{2}$$
31 $$1 - 124.T + 2.97e4T^{2}$$
37 $$1 + 56.5iT - 5.06e4T^{2}$$
41 $$1 + 135. iT - 6.89e4T^{2}$$
43 $$1 - 259.T + 7.95e4T^{2}$$
47 $$1 + 217.T + 1.03e5T^{2}$$
53 $$1 + 529. iT - 1.48e5T^{2}$$
59 $$1 - 685. iT - 2.05e5T^{2}$$
61 $$1 - 149.T + 2.26e5T^{2}$$
67 $$1 + 409.T + 3.00e5T^{2}$$
71 $$1 - 885. iT - 3.57e5T^{2}$$
73 $$1 - 269. iT - 3.89e5T^{2}$$
79 $$1 - 902. iT - 4.93e5T^{2}$$
83 $$1 - 623. iT - 5.71e5T^{2}$$
89 $$1 - 986. iT - 7.04e5T^{2}$$
97 $$1 - 179. 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

−9.694809169906433160830399931823, −8.854303388233849300012336119303, −7.995242646865038077821090107392, −6.97721119076040681799634148352, −6.23009209431378274292310386045, −5.41970062990660542028902014251, −4.17861576931128922839475592410, −3.68042106942316805562399419403, −2.53673143791863739925948912761, −1.07494646629641473226882902636, 0.25696968284748953370564800824, 1.56503722281491082712169910055, 2.65187081726142270776624743731, 3.56643333238305816229543500733, 4.74334539000873610975285682304, 5.96290106333999597588134078866, 6.31251497306415632266335184019, 7.26895241648783708684616538391, 8.124703624292191447551745621527, 9.072465341113567565334860136615