# Properties

 Label 2-507-1.1-c1-0-23 Degree $2$ Conductor $507$ Sign $-1$ Analytic cond. $4.04841$ Root an. cond. $2.01206$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.24·2-s + 3-s − 0.445·4-s − 2.80·5-s + 1.24·6-s − 4.80·7-s − 3.04·8-s + 9-s − 3.49·10-s + 1.46·11-s − 0.445·12-s − 5.98·14-s − 2.80·15-s − 2.91·16-s − 2.44·17-s + 1.24·18-s + 2.54·19-s + 1.24·20-s − 4.80·21-s + 1.82·22-s − 3.51·23-s − 3.04·24-s + 2.85·25-s + 27-s + 2.13·28-s + 1.85·29-s − 3.49·30-s + ⋯
 L(s)  = 1 + 0.881·2-s + 0.577·3-s − 0.222·4-s − 1.25·5-s + 0.509·6-s − 1.81·7-s − 1.07·8-s + 0.333·9-s − 1.10·10-s + 0.442·11-s − 0.128·12-s − 1.60·14-s − 0.723·15-s − 0.727·16-s − 0.593·17-s + 0.293·18-s + 0.583·19-s + 0.278·20-s − 1.04·21-s + 0.389·22-s − 0.733·23-s − 0.622·24-s + 0.570·25-s + 0.192·27-s + 0.403·28-s + 0.343·29-s − 0.637·30-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$507$$    =    $$3 \cdot 13^{2}$$ Sign: $-1$ Analytic conductor: $$4.04841$$ Root analytic conductor: $$2.01206$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{507} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 507,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 - T$$
13 $$1$$
good2 $$1 - 1.24T + 2T^{2}$$
5 $$1 + 2.80T + 5T^{2}$$
7 $$1 + 4.80T + 7T^{2}$$
11 $$1 - 1.46T + 11T^{2}$$
17 $$1 + 2.44T + 17T^{2}$$
19 $$1 - 2.54T + 19T^{2}$$
23 $$1 + 3.51T + 23T^{2}$$
29 $$1 - 1.85T + 29T^{2}$$
31 $$1 + 7.63T + 31T^{2}$$
37 $$1 + 4.55T + 37T^{2}$$
41 $$1 - 1.24T + 41T^{2}$$
43 $$1 - 2.38T + 43T^{2}$$
47 $$1 - 12.8T + 47T^{2}$$
53 $$1 + 8.85T + 53T^{2}$$
59 $$1 - 2.17T + 59T^{2}$$
61 $$1 + 7.82T + 61T^{2}$$
67 $$1 + 3.58T + 67T^{2}$$
71 $$1 + 8.83T + 71T^{2}$$
73 $$1 + 7.69T + 73T^{2}$$
79 $$1 + 4.02T + 79T^{2}$$
83 $$1 + 0.652T + 83T^{2}$$
89 $$1 - 6.29T + 89T^{2}$$
97 $$1 + 10.0T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$