# Properties

 Degree $2$ Conductor $503$ Sign $-1$ Motivic weight $1$ Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.17·2-s − 0.791·3-s − 0.624·4-s + 0.178·5-s − 0.928·6-s − 0.0809·7-s − 3.07·8-s − 2.37·9-s + 0.209·10-s − 4.30·11-s + 0.494·12-s + 1.10·13-s − 0.0948·14-s − 0.141·15-s − 2.35·16-s − 3.06·17-s − 2.78·18-s − 2.15·19-s − 0.111·20-s + 0.0640·21-s − 5.04·22-s + 9.15·23-s + 2.43·24-s − 4.96·25-s + 1.29·26-s + 4.25·27-s + 0.0505·28-s + ⋯
 L(s)  = 1 + 0.829·2-s − 0.457·3-s − 0.312·4-s + 0.0799·5-s − 0.379·6-s − 0.0305·7-s − 1.08·8-s − 0.791·9-s + 0.0662·10-s − 1.29·11-s + 0.142·12-s + 0.305·13-s − 0.0253·14-s − 0.0365·15-s − 0.589·16-s − 0.743·17-s − 0.655·18-s − 0.495·19-s − 0.0249·20-s + 0.0139·21-s − 1.07·22-s + 1.90·23-s + 0.497·24-s − 0.993·25-s + 0.253·26-s + 0.818·27-s + 0.00955·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$503$$ Sign: $-1$ Motivic weight: $$1$$ Character: $\chi_{503} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 503,\ (\ :1/2),\ -1)$$

## 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$$F_p(T)$
bad503 $$1 + T$$
good2 $$1 - 1.17T + 2T^{2}$$
3 $$1 + 0.791T + 3T^{2}$$
5 $$1 - 0.178T + 5T^{2}$$
7 $$1 + 0.0809T + 7T^{2}$$
11 $$1 + 4.30T + 11T^{2}$$
13 $$1 - 1.10T + 13T^{2}$$
17 $$1 + 3.06T + 17T^{2}$$
19 $$1 + 2.15T + 19T^{2}$$
23 $$1 - 9.15T + 23T^{2}$$
29 $$1 + 6.17T + 29T^{2}$$
31 $$1 + 9.82T + 31T^{2}$$
37 $$1 - 4.08T + 37T^{2}$$
41 $$1 - 4.58T + 41T^{2}$$
43 $$1 - 2.03T + 43T^{2}$$
47 $$1 + 5.12T + 47T^{2}$$
53 $$1 - 2.69T + 53T^{2}$$
59 $$1 - 9.21T + 59T^{2}$$
61 $$1 + 14.0T + 61T^{2}$$
67 $$1 - 5.62T + 67T^{2}$$
71 $$1 + 4.88T + 71T^{2}$$
73 $$1 - 10.9T + 73T^{2}$$
79 $$1 - 17.4T + 79T^{2}$$
83 $$1 + 2.48T + 83T^{2}$$
89 $$1 - 5.57T + 89T^{2}$$
97 $$1 + 13.9T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$