# Properties

 Degree $2$ Conductor $2352$ Sign $1$ Motivic weight $3$ Primitive yes Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 3·3-s − 8.16·5-s + 9·9-s − 37.8·11-s − 39.9·13-s − 24.5·15-s − 9.93·17-s − 90.4·19-s − 118.·23-s − 58.2·25-s + 27·27-s − 78.4·29-s + 92.0·31-s − 113.·33-s + 332.·37-s − 119.·39-s − 71.7·41-s + 115.·43-s − 73.5·45-s + 307.·47-s − 29.8·51-s − 403.·53-s + 308.·55-s − 271.·57-s + 593.·59-s − 333.·61-s + 326.·65-s + ⋯
 L(s)  = 1 + 0.577·3-s − 0.730·5-s + 0.333·9-s − 1.03·11-s − 0.851·13-s − 0.421·15-s − 0.141·17-s − 1.09·19-s − 1.07·23-s − 0.466·25-s + 0.192·27-s − 0.502·29-s + 0.533·31-s − 0.598·33-s + 1.47·37-s − 0.491·39-s − 0.273·41-s + 0.411·43-s − 0.243·45-s + 0.955·47-s − 0.0818·51-s − 1.04·53-s + 0.757·55-s − 0.630·57-s + 1.31·59-s − 0.699·61-s + 0.622·65-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2352$$    =    $$2^{4} \cdot 3 \cdot 7^{2}$$ Sign: $1$ Motivic weight: $$3$$ Character: $\chi_{2352} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 2352,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.224924646$$ $$L(\frac12)$$ $$\approx$$ $$1.224924646$$ $$L(\frac{5}{2})$$ 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 - 3T$$
7 $$1$$
good5 $$1 + 8.16T + 125T^{2}$$
11 $$1 + 37.8T + 1.33e3T^{2}$$
13 $$1 + 39.9T + 2.19e3T^{2}$$
17 $$1 + 9.93T + 4.91e3T^{2}$$
19 $$1 + 90.4T + 6.85e3T^{2}$$
23 $$1 + 118.T + 1.21e4T^{2}$$
29 $$1 + 78.4T + 2.43e4T^{2}$$
31 $$1 - 92.0T + 2.97e4T^{2}$$
37 $$1 - 332.T + 5.06e4T^{2}$$
41 $$1 + 71.7T + 6.89e4T^{2}$$
43 $$1 - 115.T + 7.95e4T^{2}$$
47 $$1 - 307.T + 1.03e5T^{2}$$
53 $$1 + 403.T + 1.48e5T^{2}$$
59 $$1 - 593.T + 2.05e5T^{2}$$
61 $$1 + 333.T + 2.26e5T^{2}$$
67 $$1 - 743.T + 3.00e5T^{2}$$
71 $$1 - 728.T + 3.57e5T^{2}$$
73 $$1 - 801.T + 3.89e5T^{2}$$
79 $$1 + 1.06e3T + 4.93e5T^{2}$$
83 $$1 - 906.T + 5.71e5T^{2}$$
89 $$1 + 1.11e3T + 7.04e5T^{2}$$
97 $$1 + 1.48e3T + 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

−8.338941638912948123355467752026, −7.985953179768969249606102164046, −7.33804015967586631224285422015, −6.39559029664777746351088489161, −5.41794736094286262928782943348, −4.44224226278406675913669249475, −3.85764998581098195098715083384, −2.69675777371874876366061734684, −2.07243133821239135972075228488, −0.45039596435440305849176762267, 0.45039596435440305849176762267, 2.07243133821239135972075228488, 2.69675777371874876366061734684, 3.85764998581098195098715083384, 4.44224226278406675913669249475, 5.41794736094286262928782943348, 6.39559029664777746351088489161, 7.33804015967586631224285422015, 7.985953179768969249606102164046, 8.338941638912948123355467752026