# Properties

 Label 2-507-1.1-c3-0-38 Degree $2$ Conductor $507$ Sign $-1$ Analytic cond. $29.9139$ Root an. cond. $5.46936$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 3·3-s − 8·4-s − 5.19·5-s − 10.3·7-s + 9·9-s + 51.9·11-s + 24·12-s + 15.5·15-s + 64·16-s + 117·17-s − 24.2·19-s + 41.5·20-s + 31.1·21-s − 18·23-s − 98·25-s − 27·27-s + 83.1·28-s − 99·29-s − 193.·31-s − 155.·33-s + 54·35-s − 72·36-s + 112.·37-s + 36.3·41-s + 82·43-s − 415.·44-s − 46.7·45-s + ⋯
 L(s)  = 1 − 0.577·3-s − 4-s − 0.464·5-s − 0.561·7-s + 0.333·9-s + 1.42·11-s + 0.577·12-s + 0.268·15-s + 16-s + 1.66·17-s − 0.292·19-s + 0.464·20-s + 0.323·21-s − 0.163·23-s − 0.784·25-s − 0.192·27-s + 0.561·28-s − 0.633·29-s − 1.12·31-s − 0.822·33-s + 0.260·35-s − 0.333·36-s + 0.500·37-s + 0.138·41-s + 0.290·43-s − 1.42·44-s − 0.154·45-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$507$$    =    $$3 \cdot 13^{2}$$ Sign: $-1$ Analytic conductor: $$29.9139$$ Root analytic conductor: $$5.46936$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 507,\ (\ :3/2),\ -1)$$

## 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$$F_p(T)$
bad3 $$1 + 3T$$
13 $$1$$
good2 $$1 + 8T^{2}$$
5 $$1 + 5.19T + 125T^{2}$$
7 $$1 + 10.3T + 343T^{2}$$
11 $$1 - 51.9T + 1.33e3T^{2}$$
17 $$1 - 117T + 4.91e3T^{2}$$
19 $$1 + 24.2T + 6.85e3T^{2}$$
23 $$1 + 18T + 1.21e4T^{2}$$
29 $$1 + 99T + 2.43e4T^{2}$$
31 $$1 + 193.T + 2.97e4T^{2}$$
37 $$1 - 112.T + 5.06e4T^{2}$$
41 $$1 - 36.3T + 6.89e4T^{2}$$
43 $$1 - 82T + 7.95e4T^{2}$$
47 $$1 + 72.7T + 1.03e5T^{2}$$
53 $$1 + 261T + 1.48e5T^{2}$$
59 $$1 - 789.T + 2.05e5T^{2}$$
61 $$1 + 719T + 2.26e5T^{2}$$
67 $$1 - 703.T + 3.00e5T^{2}$$
71 $$1 + 467.T + 3.57e5T^{2}$$
73 $$1 - 684.T + 3.89e5T^{2}$$
79 $$1 + 440T + 4.93e5T^{2}$$
83 $$1 + 1.19e3T + 5.71e5T^{2}$$
89 $$1 - 1.51e3T + 7.04e5T^{2}$$
97 $$1 + 1.15e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$