Properties

 Label 2-1344-56.27-c3-0-52 Degree $2$ Conductor $1344$ Sign $0.819 + 0.573i$ 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 − 12.8·5-s + (6.32 − 17.4i)7-s − 9·9-s + 25.0·11-s + 64.1·13-s + 38.4i·15-s + 77.4i·17-s − 49.9i·19-s + (−52.2 − 18.9i)21-s − 80.5i·23-s + 39.4·25-s + 27i·27-s + 159. i·29-s + 96.5·31-s + ⋯
 L(s)  = 1 − 0.577i·3-s − 1.14·5-s + (0.341 − 0.939i)7-s − 0.333·9-s + 0.686·11-s + 1.36·13-s + 0.662i·15-s + 1.10i·17-s − 0.603i·19-s + (−0.542 − 0.197i)21-s − 0.730i·23-s + 0.315·25-s + 0.192i·27-s + 1.01i·29-s + 0.559·31-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$1344$$    =    $$2^{6} \cdot 3 \cdot 7$$ Sign: $0.819 + 0.573i$ 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: $$(2,\ 1344,\ (\ :3/2),\ 0.819 + 0.573i)$$

Particular Values

 $$L(2)$$ $$\approx$$ $$1.845137725$$ $$L(\frac12)$$ $$\approx$$ $$1.845137725$$ $$L(\frac{5}{2})$$ 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 + (-6.32 + 17.4i)T$$
good5 $$1 + 12.8T + 125T^{2}$$
11 $$1 - 25.0T + 1.33e3T^{2}$$
13 $$1 - 64.1T + 2.19e3T^{2}$$
17 $$1 - 77.4iT - 4.91e3T^{2}$$
19 $$1 + 49.9iT - 6.85e3T^{2}$$
23 $$1 + 80.5iT - 1.21e4T^{2}$$
29 $$1 - 159. iT - 2.43e4T^{2}$$
31 $$1 - 96.5T + 2.97e4T^{2}$$
37 $$1 - 274. iT - 5.06e4T^{2}$$
41 $$1 - 299. iT - 6.89e4T^{2}$$
43 $$1 + 385.T + 7.95e4T^{2}$$
47 $$1 - 418.T + 1.03e5T^{2}$$
53 $$1 - 665. iT - 1.48e5T^{2}$$
59 $$1 - 445. iT - 2.05e5T^{2}$$
61 $$1 - 599.T + 2.26e5T^{2}$$
67 $$1 - 675.T + 3.00e5T^{2}$$
71 $$1 + 877. iT - 3.57e5T^{2}$$
73 $$1 + 696. iT - 3.89e5T^{2}$$
79 $$1 - 1.24e3iT - 4.93e5T^{2}$$
83 $$1 + 238. iT - 5.71e5T^{2}$$
89 $$1 + 743. iT - 7.04e5T^{2}$$
97 $$1 + 1.55e3iT - 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

−8.737901611309333570090077642315, −8.370740302696201523883682661360, −7.57005939904568873849146554491, −6.75605170951896313322328476867, −6.14366379793183340822456049849, −4.68794735131738499831297452965, −3.96491869128738761767105621002, −3.20671798270406420303743442921, −1.52118811726926010135124149449, −0.75739727061452556738260725437, 0.67227302627022618757483171474, 2.16600871255323077968343416357, 3.54628152624926621414080032343, 3.94591967845853389350549316519, 5.10516272538238260751368388183, 5.86990883471926307741109311634, 6.88806643643404344313290170163, 7.939468254325984944774576004656, 8.511573450298802520021916351274, 9.206348513229047090374724667716