# Properties

 Degree 2 Conductor 4003 Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.41·2-s − 1.41·3-s − 1.41·5-s + 2.00·6-s − 7-s + 2.82·8-s − 0.999·9-s + 2.00·10-s + 0.828·11-s + 1.17·13-s + 1.41·14-s + 2.00·15-s − 4.00·16-s + 4.65·17-s + 1.41·18-s − 4.65·19-s + 1.41·21-s − 1.17·22-s − 8.82·23-s − 4·24-s − 2.99·25-s − 1.65·26-s + 5.65·27-s − 1.17·29-s − 2.82·30-s + ⋯
 L(s)  = 1 − 1.00·2-s − 0.816·3-s − 0.632·5-s + 0.816·6-s − 0.377·7-s + 0.999·8-s − 0.333·9-s + 0.632·10-s + 0.249·11-s + 0.324·13-s + 0.377·14-s + 0.516·15-s − 1.00·16-s + 1.12·17-s + 0.333·18-s − 1.06·19-s + 0.308·21-s − 0.249·22-s − 1.84·23-s − 0.816·24-s − 0.599·25-s − 0.324·26-s + 1.08·27-s − 0.217·29-s − 0.516·30-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$4003$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{4003} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 4003,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 4003$, $F_p(T) = 1 - a_p T + p T^2 .$If $p = 4003$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad4003 $$1+O(T)$$
good2 $$1 + 1.41T + 2T^{2}$$
3 $$1 + 1.41T + 3T^{2}$$
5 $$1 + 1.41T + 5T^{2}$$
7 $$1 + T + 7T^{2}$$
11 $$1 - 0.828T + 11T^{2}$$
13 $$1 - 1.17T + 13T^{2}$$
17 $$1 - 4.65T + 17T^{2}$$
19 $$1 + 4.65T + 19T^{2}$$
23 $$1 + 8.82T + 23T^{2}$$
29 $$1 + 1.17T + 29T^{2}$$
31 $$1 - 10.4T + 31T^{2}$$
37 $$1 + 4.17T + 37T^{2}$$
41 $$1 - 5.17T + 41T^{2}$$
43 $$1 + 43T^{2}$$
47 $$1 + 3.07T + 47T^{2}$$
53 $$1 - 9.31T + 53T^{2}$$
59 $$1 + 8.82T + 59T^{2}$$
61 $$1 - 7.17T + 61T^{2}$$
67 $$1 - 12.6T + 67T^{2}$$
71 $$1 - 6T + 71T^{2}$$
73 $$1 + 3T + 73T^{2}$$
79 $$1 - 11.4T + 79T^{2}$$
83 $$1 - 17.1T + 83T^{2}$$
89 $$1 + 11.8T + 89T^{2}$$
97 $$1 - 1.75T + 97T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}