Properties

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

Related objects

Dirichlet series

 L(s)  = 1 + 3·3-s − 11.4·5-s + 9·9-s + 52.5·11-s + 5.48·13-s − 34.3·15-s − 85.3·17-s − 110.·19-s + 209.·23-s + 5.99·25-s + 27·27-s − 132.·29-s − 49.3·31-s + 157.·33-s + 160.·37-s + 16.4·39-s − 138.·41-s + 365.·43-s − 103.·45-s + 131.·47-s − 256.·51-s − 561.·53-s − 601.·55-s − 331.·57-s − 436.·59-s + 291.·61-s − 62.7·65-s + ⋯
 L(s)  = 1 + 0.577·3-s − 1.02·5-s + 0.333·9-s + 1.44·11-s + 0.116·13-s − 0.591·15-s − 1.21·17-s − 1.33·19-s + 1.89·23-s + 0.0479·25-s + 0.192·27-s − 0.850·29-s − 0.285·31-s + 0.832·33-s + 0.712·37-s + 0.0675·39-s − 0.526·41-s + 1.29·43-s − 0.341·45-s + 0.407·47-s − 0.703·51-s − 1.45·53-s − 1.47·55-s − 0.771·57-s − 0.962·59-s + 0.612·61-s − 0.119·65-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: $$1$$ Selberg data: $$(2,\ 2352,\ (\ :3/2),\ -1)$$

Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ 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 + 11.4T + 125T^{2}$$
11 $$1 - 52.5T + 1.33e3T^{2}$$
13 $$1 - 5.48T + 2.19e3T^{2}$$
17 $$1 + 85.3T + 4.91e3T^{2}$$
19 $$1 + 110.T + 6.85e3T^{2}$$
23 $$1 - 209.T + 1.21e4T^{2}$$
29 $$1 + 132.T + 2.43e4T^{2}$$
31 $$1 + 49.3T + 2.97e4T^{2}$$
37 $$1 - 160.T + 5.06e4T^{2}$$
41 $$1 + 138.T + 6.89e4T^{2}$$
43 $$1 - 365.T + 7.95e4T^{2}$$
47 $$1 - 131.T + 1.03e5T^{2}$$
53 $$1 + 561.T + 1.48e5T^{2}$$
59 $$1 + 436.T + 2.05e5T^{2}$$
61 $$1 - 291.T + 2.26e5T^{2}$$
67 $$1 - 593.T + 3.00e5T^{2}$$
71 $$1 + 775.T + 3.57e5T^{2}$$
73 $$1 + 330.T + 3.89e5T^{2}$$
79 $$1 + 243.T + 4.93e5T^{2}$$
83 $$1 - 332.T + 5.71e5T^{2}$$
89 $$1 - 979.T + 7.04e5T^{2}$$
97 $$1 + 466.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$